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Elementary Logic

Logic

1 Logic : relation of consequence between the premises and the conclusion of sound argument

1.1 Sound(an argument): if conclusion is a consequence of its premises ~ correct, valid

1.2 Unsound : not Sound

1.3 Argument : System of declarative sentences

1.3.1 Conclusion

1.3.2 Premises

1.3.3 Sentences : linguistic expression stating a complete thought

1.3.3.1 Declarative

1.3.3.2 Interrogative

1.3.3.3 Imperative

1.4 Criterion for the soundness of an argument : An argument is sound iff it is not possible for its premises to be true and its conclusion false

1.4.1 Necessary truth : a Sentence is necessary truth iff no conceivable circumstances of being false

1.4.2 Conditional, consequent : If (corresponding conditional) then (consequent)

1.4.3 A sentence is a necessary truth iff no unsound argument of which it is the conclusion

1.5 Parenthetical remark

1.5.1 If a sentence is known to be true, it is true,

1.5.1.1 Even if a sentence is true, it could not be known to be true.

1.5.2 Logically possible (a sentence) : the hypothesis it occurs is compatible with the laws of logic

1.5.3 Proposition, statement, thought, judgement

1.6 Logical form

1.6.1 If a sentence is necessary truth then all sentences of the same logical form are also necessarily true.

1.6.2 A truth of logic ( a sentence) iff it is a substitution-instance of a matrix all instances of which are necessary truths

1.6.2.1 Matrix : formal expressions built with logical words together with sentential, class, or individual letters

1.6.2.1.1 Logical words : and, or, if…then, not, all, is, etc

1.6.2.1.2 Sentential letters : sentence

1.6.2.1.3 Class letters : a set of elements

1.6.2.1.4 Individual letters : elements

1.7 Artificial language (Formalized language) : grammatically simpler, regular than natural language

Logic Preliminaries

2 Further preliminaries

2.1 Use and mention

2.1.1 Use : name (identifier)

2.1.2 Mention : use the name for reference (object which identifier indicates)

2.2 Sense and denotation

2.2.1 Sense (an expression) : meaning

2.2.2 Denotation (an expression) : objects which it refers

2.2.3 Condition of sense and denotation

2.2.3.1 The denotation (a complex expression) is a function of the denotations of its parts

2.2.3.2 The sense (a complex expression) is a function of the senses of its parts

2.2.4 Ordinary vs Oblique sense/denotation

2.2.4.1 Oblique denotation = Ordinary sense

2.2.4.2 Direct occurrence (an occurrence of a name or description in a expression) : in the context it has ordinary sense

2.3 Variables : expression of generalization, must specify substituents

2.3.1 Substituents : a set of values which could substitute a variable

2.3.2 Values : names

2.4 Sentence forms

2.4.1 Replacement of a direct occurrence of a sentence with the expression with same ordinary denotation does not change its truth value

2.4.2 Replacement of an indirect occurrence of a sentence with the expression with same ordinary denotation can change its truth value

2.4.3 Sentence form : it is an expression that is a sentence or is obtainable from a sentence by replacing some or all direct occurrences of names by variables

2.4.4 Quantifier : often asserted in sentence forms

2.4.4.1 Universal quantifier : For every x, ..

2.4.4.2 Existential quantifier : There is an x such that …

2.4.5 Variables

2.4.5.1 Free occurrences : needs additional quantifiers or replacements to obtain a sentence

2.4.5.2 Bound : already has those

2.5 Description forms

2.5.1 Replacement of a direct occurrence of a description with the expression with same ordinary denotation does not change its truth value

2.6 Set ~ Class

2.6.1 Elements : objects constituting a set ~ members

2.6.1.1 Sets having same members are identical

2.6.2 Empty set : no members

2.6.3 Universal set : set of all objects satisfying (x is identical with x)

2.6.4 원소나열법 : notation for any finite number of occurrences of variables, names, or descriptions

2.6.5 A subset of B : every element of a set A is also an element of a set B ~ included in

2.6.5.1 Φ⊂A

2.6.5.2 A⊂A

2.6.5.3 A⊂(Universal set)

2.6.5.4 If A⊂B and B⊂A, then A = B

2.6.5.5 If A⊂B and B⊂C, then A⊂C

2.6.6 The Union of A and B : a set of all members belonging to A or to B

2.6.7 The Intersection of A and B : a set of all members belonging to A and to B

2.6.8 The Complement of A : a set of all members not belonging to A

2.6.9 Russell’s Antinomy : K = {x | x is not an element of x} K ∈K or K not∈ K ?

2.6.9.1 Give up Assumption that all sets can themselves be members of sets

2.6.10 Relation : any set of ordered n-tuples of objects is an n-ary relation

2.6.10.1 Binary Relation : x R y = <x,y> ∈ R

2.6.10.2 Domain : a set of all objects x s.t. for some y , x R y

2.6.10.3 Converse domain : a set of all objects y s.t. for some x, x R y

2.6.10.4 Field (binary relation) : (Domain) U (Converse domain)

2.6.10.5 Converse (binary relation R) : for all objects x, y , x R y iff y S x

2.6.10.6 Function (a binary relation R) : for all objects x, y, z , if x R y and x R z , then y = z

2.6.10.7 1-1 relation (binary relation R) : R and its converse are both function

2.6.10.8 N-ary operation (n+1-ary relation R) : with respect to set D, for each n-tuple <x1, x2, … , xn> of objects in D there uniquely exists an object y in D s.t. <x1,x2,..,xn,y> ∈ R

2.7 Object-language and Metalanguage

2.7.1 Explain one language by using another, then former is Object-language and latter is Metalanguage (ex. artificial language explained by English)

2.7.2 Metalinguistic variables : variables of the metalanguage, Greek letters

Formalized Language L

3 Formalized Language L

3.1 Grammar of L

3.1.1 Expression of L : finite length strings of symbols

3.1.1.1 Variables : u~z with/w/o subscripts

3.1.1.1.1 Subscript : lower right Arabic numerals

3.1.1.2 Constants

3.1.1.2.1 –, V, (, ), &, \(\implies), \(\iff\), ∃

3.1.1.2.2 Non-logical constants

3.1.1.2.2.1 Predicates : capital Italic letter

3.1.1.2.2.1.1 N-ary predicate : superscript n

3.1.1.2.2.1.1.1 Superscript : upper right Arabic numerals

3.1.1.2.2.1.2 Sentential letter : no superscript

3.1.1.2.2.2 Individual constants : a ~ t

3.1.1.3 Individual symbol : variable or individual constant

3.1.1.4 Atomic formula : a sentential letter or n-ary predicate with a string of length n of individual symbols

3.1.1.4.1 General : a formula not an atomic formula, begins with a universal / existential quantifier

3.1.1.4.2 Molecular : a formula not an atomic formula, otherwise

3.1.1.5 Formula : atomic formula or built up from one or more atomic formulas by a finite number of applications as :

3.1.1.5.1 If Φ is a formula, then -Φ is a formula

3.1.1.5.1.1 Negation

3.1.1.5.2 If Φ and Ψ are formulas, then (ΦVΨ) , (Φ&Ψ), (Φ\(\implies)Ψ), (Φ\(\iff\)Ψ) are formulas

3.1.1.5.2.1 Disjunction(ΦVΨ), Conjunction(Φ&Ψ), conditional(Φ\(\implies)Ψ) ( antecedent, consequent), biconditional(Φ\(\iff\)Ψ)

3.1.1.5.3 If Φ is a formula and (α)Φ and (∃α)Φ are formulas

3.1.1.5.3.1 Universal Quantifier : (α)

3.1.1.5.3.2 Existential Quantifier : (∃α)

3.1.1.5.3.3 Universal generalization (α)Φ, Existential generalization (∃α)Φ

3.1.1.6 Occurrence of a variable α in formula Φ

3.1.1.6.1 Bound : within an occurrence in Φ of a formula of the form (α)Ψ or of the form (∃α)Ψ

3.1.1.6.2 Free occurrence

3.1.1.7 Occurrence of a formula Ψ in formula Φ

3.1.1.7.1 Bound : within an occurrence in Φ of a formula of the form (α)θ or (∃α)θ

3.1.1.7.2 Free occurrence

3.1.1.8 Order of a formula

3.1.1.8.1 Order 1 : Atomic formulas

3.1.1.8.2 Order of Disjunction, Conjunction, conditional, bidirectional is 1 + (Maximum order of two formulas)

3.1.1.8.3 Order n formula Φ \(\implies) -Φ, (α)Φ (∃α)Φ is order n+1 where α is variable

3.2 Notational conventions

3.2.1 Outermost parentheses are occasionally omitted

Interpretations and Validity

4 Interpretations and Validity

4.1 Interpretations of L : Given a sentence Φ in L assign a denotation to each non-logical constant occurring in Φ

4.1.1 Specify Universe of discourse : non-empty domain D

4.1.2 Assign to each individual constant an element of D

4.1.3 Assign to each n-ary predicate an n-ary relation among element of D

4.1.4 Assign to each sentential letter one of truth values T or F

4.2 Truth : Φ is true under I

4.2.1 Let I be any interpretation and Φ be any quantifier-free sentence of L

4.2.1.1 If Φ is sentential letter, then Φ is true under I iff I assigns T to Φ

4.2.1.2 If Φ is atomic and not a sentential , then Φ is true under I iff the objects I assigns to the individual constants of Φ are related by the relation that I assigns to the predicate of Φ

4.2.1.3 If Φ = - Ψ, then Φ is true under I iff Ψ is not true under I

4.2.1.4 If Φ = (Ψ V χ) for sentences Ψ, χ, then Φ is true under I iff Ψ is true under I or χ is true under I or both

4.2.1.5 If Φ = (Ψ & χ) for sentences Ψ, χ, then Φ is true under I iff Ψ is true under I and χ is true under I

4.2.1.6 If Φ = (Ψ \(\implies) χ) for sentences Ψ, χ, then Φ is true under I iff Ψ is not true under I or χ is true under I or both

4.2.1.7 If Φ = (Ψ \(\iff\) χ) for sentences Ψ, χ, then Φ is true under I iff either Ψ and χ are both true or both not true under I.

4.2.2 Φ is false under I iff Φ is not true under I

4.2.3 Every quantifier-free sentence is an atomic sentence or is a molecular compound of shorter sentences

4.2.4 Let I and I’ be interpretations of L, and β be an individual constant; the I is β-variant of I’ iff I and I’ are the same or differ only in what they assign to β

4.2.5 Let Φ be any sentence of L, α a variable, and β the first individual constant not occurring in Φ.

4.2.5.1 If Φ is sentential letter, then Φ is true under I iff I assigns T to Φ

4.2.5.2 If Φ is atomic and not a sentential , then Φ is true under I iff the objects I assigns to the individual constants of Φ are related by the relation that I assigns to the predicate of Φ

4.2.5.3 If Φ = - Ψ, then Φ is true under I iff Ψ is not true under I

4.2.5.4 If Φ = (Ψ V χ) for sentences Ψ, χ, then Φ is true under I iff Ψ is true under I or χ is true under I or both

4.2.5.5 If Φ = (Ψ & χ) for sentences Ψ, χ, then Φ is true under I iff Ψ is true under I and χ is true under I

4.2.5.6 If Φ = (Ψ \(\implies) χ) for sentences Ψ, χ, then Φ is true under I iff Ψ is not true under I or χ is true under I or both

4.2.5.7 If Φ = (Ψ \(\iff\) χ) for sentences Ψ, χ, then Φ is true under I iff either Ψ and χ are both true or both not true under I.

4.2.5.8 If Φ = (α)Ψ, then Φ is true under I iff Ψ α/β is true under every β-variant of I

4.2.5.8.1 α/β : replace all free occurrences of α by occurrences of some individual constant β

4.2.5.9 If Φ = (∃α)Ψ, then Φ is true under I iff Ψ α/β is true under at least one β-variant of I

4.2.6 Φ is true/false under I = I assigns the truth-value T/F to Φ

4.2.7 Complete Interpretations : for each element of the domain of I, there is an individual constant to which I assigns that element as denotation

4.2.7.1 Φ is true under all complete interpretations iff Φ is true under all interpretations

4.3 Validity, Consequence, Consistency

4.3.1 Valid (a sentence Φ) : Φ is true under every interpretation

4.3.1.1 Not valid : a nonempty set D and an assignment of appropriate entities to the non-logical constants of L which makes Φ false exists

4.3.2 Consequence of a set of sentences Γ ( a sentence Φ) : there is no interpretation under which all the sentences of Γ are true and Φ is false

4.3.3 Consistent ( a set of sentences Γ ) : there is an interpretation under which all the sentences of Γ are true

4.3.4 Deduction theorem : For any sentence Φ, Φ is a consequence of the sentences Γ together with a sentence Ψ iff (Ψ \(\implies) Φ) is a consequence of Γ alone.

Translating Natural Language into L

5 Translating Natural Language into L

5.1 Introduction

5.1.1 For any Interpretation I,

5.1.1.1 ‘Ds’ is true under I iff the object I assigns to ‘s’ belongs to the set that I assigns to ‘D’

5.1.1.2 ‘-Ds’ is true under I iff the object I assigns to ‘s’ does not belong to the set that I assigns to ‘D’

5.1.1.3 ‘(∃x)Dx’ is true under I iff some element of the domain of I belongs to the set that I assigns to ‘D’.

5.1.1.4 ‘Ds \(\implies) (∃x)Dx’ is true under I iff either the object that I assigns to ‘s’ does not belong to the set that I assigns to ‘D’ or some element of the domain of I belongs to the set that I assigns to ‘D’, or both

5.2 Interpretation and Translation

5.2.1 Given the sense, the denotation(if any) is thereby fixed, but converse does not hold.

5.3 Translating connectives and quantifiers

5.3.1 Only relative to an interpretation do the expressions of L have denotations

5.3.1.1 Futile to translate unless an interpretation is given

5.3.2 If an Interpretation is given a standard translation con be obtained.

5.3.3 No simple correspondence between the form of an ordinary sentence and its counterpart in L exists.

5.3.4 Two formal sentences equivalent in L may not be equivalently represented in Natural Language.

Tautologous Sentences

6 Tautologous Sentences

6.1 Definition of tautology

6.1.1 No atomic sentence is tautologous

6.1.2 Normal assignment

6.1.2.1 An assignment A of the truth values T and F to all the sentences of L is called normal iff for each sentence Φ of L,

6.1.2.1.1 A assigns exactly one of the truth values T/F to Φ

6.1.2.1.2 If Φ = -Ψ, then A assigns T to Φ iff A do not assign T to Ψ

6.1.2.1.3 If Φ = ( Ψ V χ ) for sentences Ψ,χ then A assigns T to Φ iff A assigns T to Ψ or T to χ or both

6.1.2.1.4 If Φ = ( Ψ & χ ) for sentences Ψ,χ then A assigns T to Φ iff A assigns T to Ψ and T to χ

6.1.2.1.5 If Φ = ( Ψ \(\implies) χ ) for sentences Ψ,χ then A assigns T to Φ iff A assigns F to Ψ or T to χ or both

6.1.2.1.6 If Φ = ( Ψ \(\iff\) χ ) for sentences Ψ,χ then A assigns T to Φ iff A assigns T to both Ψ and χ or assigns F to both

6.1.3 A sentence Φ is tautologous (is a tautology) iff it is assigned the truth value T by every normal assignment of truth values T and F to the sentences of L

6.1.3.1 Every tautologous sentence is valid

6.1.4 A sentence Φ is a tautological consequence of a set of sentences Γ iff Φ is assigned the truth value T by every normal assignment that assigns the truth value T to all sentences of Γ.

6.1.4.1 A sentence Φ is a tautological consequence of a set of sentences Γ iff

6.1.4.1.1 Γ is empty and Φ is tautologous or

6.1.4.1.2 There are sentences Ψ_1, Ψ_2, … ,Ψ_n which belongs to Γ and are such that ((…(Ψ_1 & Ψ_2)& … & Ψ_n) \(\implies) Φ) is tautologous.

6.1.5 A set of sentences Γ is truth-functionally consistent iff there is at least one normal assignment that assigns the truth value T to all members of Γ

6.1.5.1 Truth-functionally inconsistent

6.2 Tautologous SC sentences; truth tables

6.2.1 For any sentence Φ, if Φ does not contain any quantifiers, then Φ is tautologous iff Φ is valid.

6.2.1.1 For any SC sentence Φ, Φ is tautologous iff Φ is valid.

6.2.1.2 For any sentence Φ, Φ is tautologous iff there is a tautologous SC sentence Ψ s.t. Φ is substitution instance of Ψ.

6.2.1.2.1 Substitution instance of SC sentence Ψ ( sentence Φ) : result of replacing sentential letters of Ψ by sentences.

6.2.2 Truth table method

6.2.2.1 A Truth value normal assignment gives to an SC sentence is determined by truth values it gives to occurring sentential letters

6.2.2.2 Suppose an SC sentence Φ containing n distinct sentential letters, there are 2^n different ways truth values T and F can be assigned to these n letters.

6.3 Deciding whether sentences are tautologous

6.3.1 Basic truth-functional component of a sentence Ψ( a sentence Φ) : Φ is atomic or general and occurs free in Ψ at least once.

6.3.2 Associated with a sentence Ψ (a SC sentence Φ) : Φ is obtained from Ψ by putting occurrences of sentential letters for all free occurrences in Ψ of its basic truth-functional components

6.3.3 For any sentences Φ, Ψ , if Ψ is an SC sentence associated with Φ, then Φ is tautologous iff Ψ is tautologous.

6.3.3.1 Construct an SC sentence Ψ associated with Φ.

6.3.3.2 By a truth-table, test Ψ for tautologousness.

6.3.3.3 Decide whether Φ is tautologous.

6.4 Rules of derivation for SC sentences

6.4.1 SC derivation : finite sequence of consecutively numbered lines, each consisting of an SC sentence together with list of numbers. ~ proof

6.4.1.1 P (Introduction of premises) Any SC sentence may be entered on a line with the line number taken as the only premise-number

6.4.1.2 MP (Modus Ponens) Ψ may be entered on a line if Φ and (Φ \(\implies) Ψ) appear on earlier lines; as premise-numbers of the new line take all premise-numbers of those earlier lines.

6.4.1.3 MT (Modus Tollens) Φ may be entered on a line if Ψ and (-Φ \(\implies) -Ψ) appear on earlier lines ; as premise-numbers of the new line take all premise-numbers of those earlier lines.

6.4.1.4 C (Conditionalization) (Φ \(\implies)Ψ) may be entered on a line if Ψ appear on earlier lines ; as premise-numbers of the new line take all premise-numbers of those earlier lines. With the exception of any that is the line number of a line on which Φ appears.

6.4.1.5 D (Definitional interchange) if Ψ is obtained from Φ by replacing an occurrence of a sentence χ in Φ by an occurrence of a sentence to which χ is definitionally equivalent, then Ψ may be entered on a line if Φ appears on an earlier line; as premise-numbers of the new line take all premise-numbers of those earlier lines

6.4.1.5.1 (Φ V Ψ) is definitionally equivalent to (-Φ\(\implies)-Ψ)

6.4.1.5.2 (Φ & Ψ) is definitionally equivalent to -(Φ\(\implies)-Ψ)

6.4.1.5.3 (Φ \(\iff\) Ψ) is definitionally equivalent to ((Φ\(\implies)Ψ)&(Ψ\(\implies)Φ))

6.4.2 SC derivation of Φ from Γ : an SC sentence Φ appears on the last line and premises belongs to a set of SC sentences Γ

6.4.2.1 SC derivable from a set of SC sentences of Γ (an SC sentence Φ) : exists SC derivation of Φ from Γ

6.4.2.2 SC Theorem ( an SC sentence Φ) : Φ is SC derivable from (empty set of sentences)

6.4.3 Any SC sentence that is a substitution instance of a previously proved SC theorem may enter on a line of proof with the empty set of premise-numbers.

6.4.3.1 Ψ may enter on a line if Φ_1, Φ_2, ..,Φ_n appear on earlier line and the conditional (Φ_1 \(\implies) (Φ_2 \(\implies) … \(\implies) (Φ_n \(\implies) Ψ)..)) is a substitution instance of already proved SC theorem.

Rules of Inference for L

7 Rules of Inference for L

7.1 Basic rules; derivability

7.1.1 Impossibility of a decision procedure for validity

7.1.2 Impossibility of step-by-step procedure for deciding whether given sentence is a consequence of other sentences

7.1.3 Derivation : finite sequence of consecutively numbered lines with premise-numbers of the line

7.1.3.1 P (Introduction of premises) : Any sentence may be entered on a line with the line number taken as the only premise-number

7.1.3.2 T (Tautological inference) : Any sentence may enter on a line if it is a tautological consequence of a set of sentences that appear on earlier lines; as premise-numbers of the new line take all premise-numbers of those earlier lines

7.1.3.3 C (Conditionalization) : (Φ \(\implies)Ψ) may be entered on a line if Ψ appear on earlier lines ; as premise-numbers of the new line take all premise-numbers of those earlier lines. With the exception of any that is the line number of a line on which Φ appears.

7.1.3.4 US (Universal specification) The sentence Φ α/β may enter on a line if (α)Φ appears on an earlier line; as premise-numbers of the new line take all premise-numbers of those earlier lines

7.1.3.5 UG (Universal generalization) The sentence (α)Φ may enter on a line if Φ α/β appears on an earlier line and β occurs neither in Φ nor in any premise of that earlier line; as premise-numbers of the new line take all premise-numbers of those earlier lines

7.1.3.6 E (Existential quantification) The sentence (∃α)Φ may enter on a line if –(α)-Φ appears on an earlier line or vice versa; as premise-numbers of the new line take all premise-numbers of those earlier lines

7.1.4 Derivation of Φ from Γ : a derivation in which a sentence Φ appears on the last line and all premises of that line belong to a set of sentences Γ

7.1.4.1 derivable from the set of sentences Γ ( a sentence Φ ) : there is a derivation of Φ from Γ

7.2 The short-cut rules EG, ES and Q

7.2.1 EG (Existential generalization) : The sentence (∃α)Φ may enter on a line if Φ α/β appears on an earlier line; as premise-numbers of the new line take all premise-numbers of those earlier lines

7.2.2 ES (Existential specification) : Suppose (∃α)Ψ appears on line I of a derivation, that Φ α/β appears as a premise on a later line j, and that Φ appears on a still later line k; and suppose further that the constant β occurs neither in Φ,Ψ, nor in any premise of line k other than Φ α/β ; than may enter on a new line. As premise-numbers of the new line take all those of lines I and k, except the number j.

7.2.3 Q (Quantifier exchange) The sentence –(∃α)Φ may enter on a line if (α)-Φ appears on earlier line, or vice versa; similarly for the pairs {(∃α)-Φ, -(α)Φ },{–(∃α)-Φ ,(α)Φ },{–(∃α)Φ , -(α)-Φ }; as premise-numbers of the new line take all premise-numbers of those earlier lines.

7.3 Theorems of logic

7.3.1 A Theorem of logic (a sentence Φ) : Φ is derivable from the empty set of sentences ~ theorem

7.3.1.1 A sentence Φ is derivable from a set of sentences Γ iff Φ is consequence of Γ

7.3.1.2 Φ is a theorem of logic iff Φ is valid.

Some Metatheorems

8 Some Metatheorems

8.1 Replacement, negation and duality, prenex normal form

8.1.1 Φ^{α_1, α_2, …,α_n}_{β_1,β_2, …,β_n} : For any formula Φ, result of replacing all free occurrences of distinct variables (α_1, α_2, …,α_n) by occurrences of individual symbols (β_1,β_2, …,β_n) respectively.

8.1.2 String of quantifiers : a quantifier | result of prefixing a quantifier to a shorter string of quantifiers

8.1.2.1 A string of universal quantifiers : a universal quantifier | result of prefixing a universal quantifier to a shorter string of quantifiers

8.1.2.2 A string of existential quantifiers : an existential quantifier | result of prefixing an existential quantifier to a shorter string of quantifiers

8.1.3 Closure of formula Ψ (a formula Φ) : Φ is sentence and Φ = Ψ or Φ is the result of prefixing a string of universal quantifiers to Ψ

8.1.4 \(\Vdash\)Φ : every closure of Φ is a theorem of logic

8.1.5 ( I )If Φ^{α_1, α_2, …,α_n}_{β_1,β_2, …,β_n} is a theorem of logic (where β_1,β_2, …,β_n are distinct individual constants not occurring in Φ) , then\(\Vdash\)Φ

8.1.5.1 Pf) UG

8.1.6 ( II )If one closure of Φ is a theorem of logic, then \(\Vdash\)Φ

8.1.6.1 Pf) US

8.1.7 ( III )If \(\Vdash\)Φ and \(\Vdash\)Φ \(\implies) Ψ, then \(\Vdash\)Ψ

8.1.7.1 Pf) US

8.1.8 ( IV )Generalization

8.1.8.1 \(\Vdash\)(α)(α’)Φ \(\iff\) (α)(α’)Φ

8.1.8.2 \(\Vdash\)(∃α)(∃α’)Φ \(\iff\) (∃α)(∃α’)Φ

8.1.8.3 \(\Vdash\)(α)(Φ & Ψ) \(\iff\) (Φ & (α)Ψ) if α does not occur free in Φ

8.1.8.4 \(\Vdash\)(α)(∃α’)(Φ & Ψ) \(\iff\) ((α)Φ & (∃α’)Ψ) if α does not occur free in Φ

8.1.9 ( V ) If \(\Vdash\)Ψ\(\iff\)Ψ’, then

8.1.9.1 \(\Vdash\)-Ψ\(\iff\)-Ψ

8.1.9.2 \(\Vdash\)(Ψ & χ) \(\iff\) (Ψ’ & χ)

8.1.9.3 \(\Vdash\)( χ & Ψ ) \(\iff\) ( χ & Ψ’ )

8.1.9.4 \(\Vdash\) ( Ψ V χ ) \(\iff\) ( Ψ’ V χ )

8.1.9.5 \(\Vdash\) ( χ V Ψ ) \(\iff\) ( χ V Ψ’ )

8.1.9.6 \(\Vdash\) (Ψ \(\implies) χ) \(\iff\) (Ψ’\(\implies)χ)

8.1.9.7 \(\Vdash\) (χ \(\implies) Ψ) \(\iff\) (χ \(\implies)Ψ’)

8.1.9.8 \(\Vdash\) (Ψ \(\iff\) χ) \(\iff\) (Ψ’ \(\iff\) χ)

8.1.9.9 \(\Vdash\) (χ \(\iff\) Ψ) \(\iff\) ( χ \(\iff\) Ψ’)

8.1.9.10 \(\Vdash\) (α)Ψ \(\iff\) (α)Ψ’

8.1.9.11 \(\Vdash\) (∃α)Ψ \(\iff\) (∃α)Ψ

8.1.9.11.1 pf) I, III, IV

8.1.10 (Replacement)

8.1.10.1 Suppose that Φ’ is like Φ except for containing an occurrence of Ψ’ where Φ contains an occurrence of Ψ, and suppose that \(\Vdash\)Ψ\(\iff\)Ψ’. Then \(\Vdash\)Φ\(\iff\)Φ’, and \(\Vdash\)Φ iff \(\Vdash\)Φ’

8.1.10.1.1 Pf) V, I, III

8.1.11 (Definition of equivalence) : for any formulas Φ and Ψ, Φ is equivalent to Ψ iff \(\Vdash\)Φ\(\iff\)Ψ

8.1.12 (Rewriting of bound variables) if formulas (α)Φ and (α’)Φ’ are alike except that former has occurrences of α where and only where the latter has occurrences of α’, then they are equivalent. Similarly for (∃α)Φ and (∃α’)Φ’.

8.1.12.1 Pf) I

8.1.13 (Negation theorem) Suppose that a formula Φ contains no occurrences of \(\implies) and \(\iff\) and that Φ’ is obtained from Φ by exchanging & and v, exchanging universal and the corresponding existential quantifiers, and by repacing atomic formulas by their negations. Then Φ’ is equivalent to -Φ.

8.1.14 (Definition of Prenex normal form) ( a formula Φ ) : Φ is either quantifier-free or consists of a string of quantifiers followed by a quantifier-free formula.

8.1.15 (Prenex normal form) : For any formula Φ there is an equivalent formula Ψ that is in prenex normal form.

8.1.15.1 Pf) reduction to prenex normal form, (Replacement), (Negation theorem), IV

8.1.15.2 If a sentence Φ has a prenex normal form in the prefix of which no existential quantifier precedes any universal quantifier, then the validity of Φ is decidable.

8.1.16 (X) if \(\Vdash\)Φ and if Ψ is the result of replacing all atomic formulas in Φ by their negations, then \(\Vdash\)Ψ

8.1.17 (Duality) if \(\Vdash\)Φ\(\iff\)Ψ and neither \(\implies) nor \(\iff\) occurs in Φ and Ψ, and if Φ* and Ψ* are obtained from Φ and Ψ, respectively, by exchanging & and v, and universal and the corresponding existential quantifiers, then \(\Vdash\)Φ\(\iff\)Ψ ; similarly if \(\Vdash\)Φ\(\iff\)Ψ, then \(\Vdash\)Ψ* \(\implies) Φ*

8.1.17.1 Pf) (Replacement), (Negation theorem), (X)

8.2 Soundness and consistency

8.2.1 Sound ( a system of inference rules) : any conclusion derived by their use will be a consequence of the premises from which it is obtained

8.2.1.1 Assertion

8.2.1.1.1 Any sentence appearing on the first line of a derivation is a consequence of the premises of that line

8.2.1.1.2 Any sentence appearing on the latter line is a consequence of its premises if all sentences appearing on earlier lines are consequence of theirs

8.2.1.2 Any sentence appearing on a line of a derivation is a consequence of the premises of that line

8.2.1.3 If a sentence Φ is derivable from a set of sentences Γ, then Φ is a consequence of Γ.

8.2.2 Consistent ( a system of inference rules ) : there is no sentence Φ such that both Φ and -Φ are derivable from (Universal set)

8.2.2.1 Transform T(Φ) (a sentence Φ) : SC sentence which is the result of deleting all individual symbols, quantifiers, and predicate superscripts from Φ

8.3 Completeness

8.3.1 Complete ( a system of inference rules ) : One can derive, from any given set of sentences, any consequence of that set

8.3.1.1 Consistent with respect to derivability ( a set of sentences Γ ) : the sentence ‘P&-P’ is not derivable from Γ ~ d-consistent

8.3.1.1.1 For any sentence Φ and set of sentences Γ, Φ is derivable form Γ iff ΓU{-Φ} is not d-consistent

8.3.1.1.2 For any set of sentences Γ, Γ is d-consistent iff at least one sentence Φ from L is not derivable from Γ

8.3.1.2 Maximal d-consistent (a set of sentences Γ) iff Γ is d-consistent and not properly included in any d-consistent set Δ.

8.3.1.2.1 Φ€Δ iff -Φ !∈ Δ

8.3.1.2.2 Φ∈Δ iff Φ is derivable from Δ

8.3.1.2.3 (Φ v Ψ) ∈ Δ iff Φ ∈ Δ or Ψ ∈ Δ

8.3.1.2.4 (Φ & Ψ) ∈ Δ iff Φ ∈ Δ and Ψ ∈ Δ

8.3.1.2.5 (Φ \(\implies) Ψ) ∈ Δ iff Φ !∈ Δ or Ψ ∈ Δ, or both

8.3.1.2.6 (Φ \(\iff\) Ψ) ∈ Δ iff Φ, Ψ ∈ Δ or Φ, Ψ !∈ Δ

8.3.1.3 ω-complete ( a set of sentences Γ ) : for every formula Φ and variable α, if (∃α)Φ belongs to Γ, then there is an individual constant β s.t. Φ α/β also belongs to Γ.

8.3.1.4 Suppose Δ is maximal d-consistent and ω-complete, then

8.3.1.4.1 (α)Φ ∈ Δ iff for every individual constant β, Φ α/β ∈ Δ

8.3.1.4.2 (∃α)Φ ∈ Δ iff for some individual constant β, Φ α/β ∈ Δ

8.3.2 Principle lemma for completeness

8.3.2.1 ( I ) for any set of sentences Γ, if Γ is d-consistent, then it is consistent.

8.3.2.1.1 ( I’ ) for any set of sentences Γ, if Γ is d-consistent and all indices of individual constants occurring in the sentences of Γ are even, then it is consistent.

8.3.2.1.2 ( II ) For any set of sentences Γ, if Γ satisfies the hypothesis of I’, then there is a set of sentences Δ that includes Γ and is maximal d-consistent and ω-complete.

8.3.2.1.2.1 Assume that all the sentences of L can be arranged in an infinite list Φ_1, Φ_2, … Φ_n with following properties.

8.3.2.1.2.1.1 Each sentence of L occurs at least one in the list

8.3.2.1.2.1.2 For each sentence of the form (∃α)Φ, there is at least one I such that Φ_i = (∃α)Φ and Φ_(i+1) = Φ α/β, where β is a ‘new’ individual constant

8.3.2.1.3 ( III ) every maximal d-consistent, ω-complete set of sentences is consistent.

8.3.2.1.3.1 Corollary : every maximal d-consistent ,ω-complete set of sentences is satisfiable by an interpretation having a domain equinumerous with the positive integers. ( denumerably infinite domain)

8.3.2.1.4 Corollary of ( I ) : For any set of sentences Γ, if Γ is d-consistent, then it is satisfiable by an interpretation having a denumerably infinite domain.

8.3.2.1.5 (Löwenheim-Skolem theorem) : if Γ is a consistent set of sentences, then Γ is satisfiable by an interpretation having a denumerably infinite domain.

8.4 A proof procedure for valid sentences

8.4.1 If a given sentence is valid, then there exists procedures which generates a proof of that sentence.

8.4.1.1 Given an arbitrary valid sentence Φ, we can reduce Φ to a sentence Ψ in prenex normal form.

8.4.1.2 Procedure ( a valid sentence Φ in prenex normal form ) :

8.4.1.2.1 Enter -Φ on the first line as a premise.

8.4.1.2.2 Derive a prenex normal form Ψ from -Φ.

8.4.1.2.3 Construct a sequence of lines satisfying the following two conditions, until a truth-functional inconsistency appears:

8.4.1.2.3.1 Whenever any universal generalization (α)θ appears on a line, particular instances θ α/β (for all individual constants β occurring in the sequence, and in any case for at least one individual constant β) shall appear on later lines, inferred by US.

8.4.1.2.3.2 Whenever any existential generalization (∃α)θ appears on a line, θ α/β (for some new individual constant β ) shall appear as a premise on a later line.

8.4.1.2.4 When a truth-functional inconsistency appears, derive ‘P&-P’ by rule T, apply ES to transfer dependence to -Φ, conditionate to obtain the theorem -Φ \(\implies) (P&-P), and apply T to obtain the theorem Φ.

Identity and Terms

9 Identity and Terms

9.1 Identity; the language L_{1}

9.1.1 Leibniz’s Law : if two things are identical, then whatever is true of the one is true of the other.

9.1.2 I^{2}_{1} : a logical constant that stands for the relation of identity in the domain of the relevant interpretation

9.1.3 L_{1} : first order predicate calculus with identity

9.1.3.1 Interpretation for the language L_{1} : nonempty domain D together with an assignment that associates with each individual constant of L_{1} an element of D, with each n-ary predicate other than I^{2}{1} an n-ary relation among elements of D, with the predicate I^{2}{1} the identity relation among elements of D, and with each sentential letter of L_{1} one of truth values T or F.

9.1.3.1.1 Every Interpretation of L_{1} is an interpretation of L, but not for converse.

9.1.3.2 Derivation in L_{1} : finite sequence of consecutively numbered lines each consisting of a sentence of L_{1} together with a set of premise-numbers.

9.1.3.2.1 Rules : P,T,C,US,UG,E and..

9.1.3.2.2 The sentence β = β may be entered on a line, with the empty set of premise-numbers

9.1.3.2.3 If a sentence Φ is like a sentence Ψ except that β and γ have been exchanged at one or more places, then Φ may be entered on a line if Ψ and β = γ appear on earlier lines; as premise-numbers of the new line take those of the earlier lines.

9.1.3.3 Derivation of Φ from Γ : a sentence Φ appears on the last line and all premises of that line belong to a set of sentences Γ

9.1.3.4 Derivable from a set of sentences Γ (a sentence Φ) : there is a derivation of Φ from Γ.

9.1.3.5 Φ is derivable from Γ iff Φ is a consequence of Γ.

9.1.3.6 Class : Division of all the individual constants of L_{1} s.t. constants α, β are assigned to the same class iff the sentence α = β is in Δ.

9.1.3.6.1 [(metalinguistic variable)] = the class to which (metalinguistic variable) belongs

9.1.3.7 8.( III )

9.1.3.8 Completeness of the predicate calculus with identity

9.1.3.9 (Löwenheim-Skolem theorem of L_{1}) : if Γ is a consistent set of sets of L_{1}, then Γ is satisfiable by an interpretation having a finite or denumerably infinite domain.

9.1.3.10 A theorem of Logic (a sentence Φ) : Φ is derivable from the empty set of sentences.

9.2 Parenthetical remark

9.2.1 Frege’s solution about identity : The truth of an identity sentence requires only that the two terms have same denotation

9.2.2 Objectionable feature

9.2.2.1 The identity relation among the element of one domain will be different from that among the elements of another

9.2.2.2 Identity relation cannot be a member of themselves.

9.3 Terms ; the Language L’

9.3.1 Operation symbol : new kind of symbol ~ functor, function sign

9.3.2 L’ : first order predicate calculus with identity and operation symbols

9.3.2.1 Result of adding operation symbols to L_{1}.

9.3.2.2 Symbol classification

9.3.2.2.1 Variables

9.3.2.2.2 Constants

9.3.2.2.2.1 Logical constants U { I^{2}_{1} }

9.3.2.2.2.2 Non-logical constants

9.3.2.2.2.2.1 Predicates – { I^{2}_{1} }

9.3.2.2.2.2.2 Operation symbols : lower-case italic letters ‘a’ through ‘t’ with or without numerical subscripts and superscripts

9.3.2.2.2.2.2.1 Individual constant : operation symbol without superscript

9.3.2.2.2.2.2.2 N-ary operation symbol : operation symbol having as superscript a numeral for the positive integer n.

9.3.2.3 N-ary predicate, sentential letter, individual symbol, formula ,sentence, bound/free occurrences are equally defined

9.3.2.4 Term : individual symbol or built up from individual and operation symbol by a finite number of following rules

9.3.2.4.1 If τ_{1}, τ_{2}, …, τ_{n} are terms and θ is an operation symbol of degree n, then θτ_{1}τ_{2}…τ_{n} is a term.

9.3.2.5 Atomic formula : expression either a sentential letter or the form πτ_{1}τ_{2}…τ_{n}, where π is n-ary predicate and τ_{1},τ_{2},…,τ_{n} are terms.

9.3.3 Interpretation of L’ : non-empty domain D together with an assignment that associates

9.3.3.1 Each individual constant of L’ an element of D

9.3.3.2 Each n-ary operation symbol an n-ary operation with respect to D

9.3.3.3 Each sentential letter of L’ one of truth values T or F

9.3.3.4 Each n-ary predicate an n-ary relation among elements of D

9.3.3.5 The binary predicate I^{2}_{1} the identity relation among elements of D

9.3.4 the value under an interpretation (a constant term τ)

9.3.4.1 if τ is an individual constant, the value of τ under I is the element of D which I assign to τ

9.3.4.2 if τ = θτ_{1}τ_{2}…τ_{n}, where τ_{1}, τ_{2}, …, τ_{n} are terms and θ is an operation symbol of degree n, then the value of the function I (θ)

9.3.5 To define ‘True under I’ we need only one change the definition given for L and L_{1}

9.3.5.1 If Φ is atomic and not a sentential letter, the Φ is true under I iff the values under I of the (constant) terms of Φ are related by the relation that I assigns to the predicate of Φ.

9.3.6 Notion valid, consequence, consistet, normal assignment, tautologous, tautological consequence, truth-functionally consistent, Derivation, derivation of Φ from Γ, derivable are all same.

9.3.7 Amended US, I from inference rules. It is complete.

Axioms for L_{1}

10 Axioms for L_{1}

10.1 Introduction

10.1.1 Axioms : easily recognizable valid sentences

10.1.2 (Modus Ponens) : A sentence Φ follows from sentences Ψ and χ by modus ponens iff χ = (Ψ -> Φ)

10.1.3 Definitionally equivalent to a formula Ψ ( a formula Φ ) : formulas χ, θ, Φ_{1}, Φ_{2} s.t. Φ and Ψ are alike except that one contains an occurrence of χ at some place where the other contains an occurrence of θ, and either

10.1.3.1 Χ = (Φ_{1} v Φ_{2}) and θ = (-Φ_{1} -> Φ_{2})

10.1.3.2 Χ = (Φ_{1} & Φ_{2}) and θ = -(Φ_{1} -> -Φ_{2})

10.1.3.3 Χ = (Φ_{1} \(\iff\) Φ_{2}) and θ = ((Φ_{1}-> Φ_{2}) & (Φ_{2}-> Φ_{1}))

10.1.3.4 Χ = (∃α)Φ_{1} and θ = -(α)-Φ_{1}

10.1.4 Proof : a finite sequence of sentences, each of which is an axiom or is definitionally equivalent to an earlier sentence of the sequence or follows from earlier sentences by modus ponens.

10.1.5 Theorem (a sentence Φ) : Φ is the last line of a proof.

10.1.5.1 \(\vdash\) Φ : all closures of Φ are theorems.

10.1.6 Notation : Φ, Ψ, χ are formulas. α is variables. β γ are individual symbols. P, Q, R, T are expressions.

10.1.7 If Φ and Φ ->Ψ are theorems, then Ψ is a theorem.

10.1.8 If Φ is a sentence, Φ->Φ is a theorem.

10.1.9 If Φ, Ψ, χ are sentences, (Ψ -> χ)-> ((Φ->Ψ) ->(Φ->χ)) is a theorem.

10.1.10 If Φ->Ψ and Ψ->χ are theorems, then Φ->χ is a theorem.

10.1.11 (Mathematical induction) : All the positive integers have some given property P ?

10.1.11.1 (weak induction)

10.1.11.1.1 1 has the property P

10.1.11.1.2 For every positive integer k, if k has the property P, then k+1 has the property P

10.1.11.2 (Strong induction)

10.1.11.2.1 For each positive integer k, if all positive integers less than k have the property P, then k has the property P.

10.1.12 Q(Φ->Ψ) ->(QΦ->QΨ) is a theorem, if Q is a string of universal quantifiers containing every variable that occurs free in Φ or Ψ.

10.1.13 If QΦ and Q(Φ->Ψ) are theorems, then QΨ is a theorem, where Q is a string of universal quantifiers.

10.1.14 \(\vdash\)(Ψ->χ)->((Φ->Ψ)->(Φ->χ))

10.1.15 If Q(Φ->Ψ) and Q(Ψ->χ) are theorems, then Q(Φ->χ) is a theorem.

10.1.16 \(\vdash\)Q(Φ->Ψ) ->(QΦ->QΨ)

10.1.17 \(\vdash\)QΦ->Φ

10.1.18 \(\vdash\)Φ->QΦ, if no variable in Q occurs free in Φ.

10.1.19 \(\vdash\)QΦ->PΦ, if every variable in P is either in Q or does not occur free in Φ.

10.1.20 If Ψ and χ are closures of Φ, then Ψ->χ is a theorem, and χ is a theorem if Ψ is a theorem.

10.1.21 If \(\vdash\) Φ and \(\vdash\)Φ->Ψ, then \(\vdash\)Ψ.

10.1.22 If \(\vdash\)Φ and Ψ is definitionally equivalent to Φ, then \(\vdash\)Ψ.

10.1.23 If \(\vdash\)Φ->Ψ and \(\vdash\)Ψ->χ, then \(\vdash\)Φ->χ

10.1.24 \(\vdash\)Φ iff \(\vdash\)QΦ->QΨ

10.1.25 If \(\vdash\)Φ->Ψ, then \(\vdash\)QΦ->QΨ

10.2 Sentential calculus (lo\(\Vvdash\) bo\(\Vvdash\) ,full of one-liner theorems)

10.2.1 (Replacement) : If \(\vdash\)Φ\(\iff\)Ψ and χ is like θ except for having an occurrence of Φ at some place where θ has an occurrence of Ψ, then \(\vdash\)χ\(\iff\)θ and \(\vdash\)χ iff \(\vdash\)θ.

10.3 Quantification theory (lo\(\Vvdash\) bo\(\Vvdash\), full of one-liner theorems)

10.4 Identity, further quantification theory, and substitution

10.4.1 (Substitution) If Φ is a theorem and Ψ is a substitution-instance of Φ, then Ψ is a theorem.

10.4.1.1 Stencil : an expression either a sentence of L_{1} or is obtainable from such a sentence by putting the counters ①, ②, ③, .. where that sentence contains occurrences of individual constants.

10.4.1.2 Substitute σ for θ in Φ ( σ : stencil of degree n, θ : n-ary predicate other than I^{2}{1}, Φ : a sentence in L{1} ) : for each occurrence of θδ_{1}δ_{2}..δ_{n} in Φ where δ_{1},δ_{2},..,δ_{n} are individual symbols, put σ(δ_{1},δ_{2},..,δ_{n})

10.4.1.3 Legitimate ( a substitution) : If in this substitution we have at all places obeyed the rule that no variable shall occur bound at any place in any substituted σ(δ_{1},δ_{2},..,δ_{n}) unless it already occurs bound at that place in σ

10.4.1.4 A substitution-instance of a sentence Φ ( a sentence Ψ) : Ψ can be obtained from Φ by a legitimate substitution.

10.5 Proofs from assumptions

10.5.1 Proof of the sentence Φ from the assumptions Γ (a set of sentences Γ) : Φ is the last term of the sequence and each term of the sequence is either an axiom or one of the assumptions Γ or follows from earlier terms by modus ponens or definitional interchange.

10.5.2 Γ\(\Vvdash\)Φ : there is a proof of Φ from the assumptions Γ

10.5.3 (empty set) \(\Vvdash\) Φ iff Φ is a theorem

10.5.4 If Φ ∈ Γ, then Γ \(\Vvdash\) Φ

10.5.5 If Γ \(\Vvdash\) Φ and Γ ⊂ Δ, then Δ \(\Vvdash\) Φ

10.5.6 If Γ \(\Vvdash\) Φ and Δ \(\Vvdash\) Φ -> Ψ, then Γ U Δ \(\Vvdash\) Ψ

10.5.7 (Deduction Theorem) Γ U { Φ } \(\Vvdash\) Ψ iff Γ \(\Vvdash\) Φ -> Ψ

10.5.8 Γ \(\Vvdash\) Φ iff Γ U {-Φ} \(\Vvdash\) ‘P&-P’

10.5.9 Γ \(\Vvdash\) Φ iff for every sentence Ψ, Γ U { - Φ } \(\Vvdash\) Ψ

10.5.10 Γ \(\Vvdash\) Φ iff Φ is a consequence of Γ.

Formalized Theory

11 Formalized Theory

11.1 Elementary theory : a theory formalized by means of elementary logic

11.2 Introduction

11.2.1 Deductive theory T formalized in the first order predicate calculus : a pair of sets T = <Δ,Γ> where Δ is a set of non-logical constants of L containing at least one predicate of degree ≥ 1 and Γ is a set of sentences of L satisfying the conditions

11.2.1.1 All non-logical constants occurring in members of Γ are members of Δ

11.2.1.2 Deductively closed : Every sentence of T that is a consequence of Γ is a member of Γ

11.2.2 Non-logical Vocabulary of T = <Δ,Γ> : Δ

11.2.3 Assertions, Theses of T = <Δ,Γ> : Γ

11.2.3.1 Deductive theory is uniquely determined by its non-logical vocabulary and its assertions.

11.2.4 Deductive theory T formalized in the first order predicate calculus with identity ( or with identity and operation symbols) : Similar, but now referring to L_{1} (or L’)

11.2.5 Consistent ( a theory T) : no sentence and its negation are both asserted by T

11.2.6 Complete ( a theory T) : For any sentence of T, either that sentence or its negation is asserted by T.

11.2.6.1 Every inconsistent theory is automatically complete

11.2.7 Independent (a set of sentences Γ) : no element Φ of Γ ~ {Φ} (special notation ~)

11.2.8 Decidable ( a set of sentences Γ) : there is a step-by-step procedure which, applied to an arbitrary expression Φ, will decide in a finite number of steps whether or not Φ belongs to Γ.

11.2.9 Axiomatizable ( a theory T) : among the assertions of a given theory T there is a decidable subset ① of which all the others are consequences

11.2.9.1 Set of axioms of T : ①

11.2.9.2 Axiomatic theory : T

11.2.10 Heteronomous system : an axiom system proposed next and judged first on the basis of whether or not the resulting theorems coincide with the assertions

11.2.11 Autonomous : axiom system proposed first and the theses of the theory are defined as those sentences of the theory following from the axioms

11.2.12 Evaluation of axiom system

11.2.12.1 Consistency

11.2.12.1.1 Semantically : give an interpretation under which all the axioms are true

11.2.12.1.2 Syntactically : without reference to interpretations, there is no sentence Φ s.t. both Φ and -Φ are derivable from the axioms

11.2.12.2 Complete : theory from an axiom system is complete

11.2.12.3 Independence : give each axiom an interpretation under which it is false but the remaining axioms are true

11.2.12.4 Categorical : all its models are isomorphic

11.2.12.4.1 Any consistent axiom system has infinitely many models.

11.2.12.4.2 (Isomorphic model) Given a set of sentences Γ of L’, interpretation I_{1} and I_{2} are isomorphic models of Γ iff

11.2.12.4.2.1 I_{1} and I_{2} are models of Γ

11.2.12.4.2.2 There is a 1-1 relation between the domains of I_{1} and I_{2} associating with each element e of the domain of I_{1} exactly one element e* of the domain of I_{2}

11.2.12.4.2.3 For each sentential letter Φ occurring in Γ, I_{1}(Φ) = I_{2}(Φ)

11.2.12.4.2.4 For each individual constant β occurring in Γ, I_{2}(β) = I_{1}(β)*

11.2.12.4.2.5 For each n-ary predicate θ occurring in Γ, the n-tuple <t_{1}, t_{2}, …, t_{n}> ∈ I_{1}(θ) iff <t_{1}, t_{2}, .., t_{n}*> ∈ I_{2}(θ), for all elements t_{1}, t_{2}, .. , t_{n} of the domain of I_{1}

11.2.12.4.2.6 For each n-ary operation θ occurring in Γ, the n-tuple <t_{1}, t_{2}, …, t_{n}, η> ∈ I_{1}(θ) iff <t_{1}, t_{2}, .., t_{n}, η> ∈ I_{2}(θ), for all elements t_{1}, t_{2}, .. , t_{n}, η of the domain of I_{1}

11.2.12.4.3 If there are models with different cardinality, the theory is not categorical.

11.2.12.4.4 Representation theorem : While not all models of the theory of groups are isomorphic to one another, every such model is isomorphic to a group of transformations.

11.3 Aristotelian syllogistic

11.3.1 T1

11.3.1.1 Non-logical vocabulary of T1 : binary predicates A^{2}, E^{2}, I^{2}, O^{2}

11.3.1.2 Axioms of assertions

11.3.1.2.1 (x)(y)(z)((A_{yz}&A_{xy})\(\implies)A_{xz}) (Barbara)

11.3.1.2.2 (x)(y)(z)((E_{yz}&A_{xy})\(\implies)E_{xz}) (Celarent)

11.3.1.2.3 (x)(y)(I_{xy} \(\implies) I_{yx}) (Conversion of ‘I’)

11.3.1.2.4 (x)(y)(E_{xy} \(\implies) E_{yx}) (Conversion of ‘E’)

11.3.1.2.5 (x)(y)(A_{xy} \(\implies) I_{yx}) (Conversion of ‘A’)

11.3.1.2.6 (x)(y)(E_{xy} \(\implies) -I_{yx}) (Definition of ‘E’)

11.3.1.2.7 (x)(y)(O_{xy} \(\implies) -A_{yx}) (Definition of ‘O’)

11.3.1.3 Evaluation of T1

11.3.1.3.1 T1 is consistent

11.3.1.3.2 T1 is not complete

11.3.1.3.3 T1 is not independent

11.3.1.3.4 T1 is not categorical

11.3.1.4 TH : Brevity convention

11.3.1.4.1 Any instance of an axiom or previously proved theorem may be entered on a line, with the empty set of premise numbers

11.3.1.4.2 Ψ may enter the line if Φ_{1}, …, Φ_{n} appear on earlier lines and Ψ is a tautological consequence of { Φ_{1}, …, Φ_{n} } and an instance of an axiom or previously proved theorem; as premise numbers of the new line take all premise-numbers of those earlier lines.

11.4 The theory of betweenness

11.4.1 T2 : axiomatized theory formulated in the first order predicate calculus with identity

11.4.1.1 Non-logical vocabulary : B^{3} (ternary)

11.4.1.2 Assertions : sentences that are true under every interpretation I satisfying :

11.4.1.2.1 The domain of I is the set of all points on some (Euclidean) straight line

11.4.1.2.2 I assigns to B^{3} the relation of betweenness “② is between ① and ③” Among points on this straight line.

11.4.1.3 Evaluation of T2

11.4.1.3.1 T2 is consistent

11.4.1.3.2 T2 is complete

11.4.1.4 Axioms of assertions

11.4.1.4.1 (x)(y)(Bxyx \(\implies) x = y)

11.4.1.4.2 (x)(y)(z)(u)((Bxyu & Byzu) \(\implies) Bxyz)

11.4.1.4.3 (x)(y)(z)(u)(((Bxyz & Byzu) & y != z) \(\implies) Bxyu)

11.4.1.4.4 (x)(y)(z)((Bxyz v Bxzy ) b (Bzxy)

11.4.1.4.5 (∃x)(∃y)x != y

11.4.1.4.6 (x)(y) (x != y \(\implies) (∃z)(Bxyz & y != z))

11.4.1.4.7 (x)(y)(x != y \(\implies) (∃z)(Bxzy & (z != x & z != y)))

11.5 Groups; Boolean algebras

11.5.1 T3 : Theory of groups formalized in the language L’

11.5.1.1 Non-logical vocabulary of T3 :

11.5.1.1.1 (Group addition) binary operation symbol f^{2}

11.5.1.1.2 (Group complementation) : singulary operation symbol f^{1}

11.5.1.1.3 (identity element) : individual constant e

11.5.1.2 Assertions

11.5.1.2.1 Conventions : Suppose τ, τ’ are terms or the result of previous application of these conventions

11.5.1.2.1.1 (τ + τ’) \(\Longleftarrow) f^{2}ττ’

11.5.1.2.1.2 Τ* for f^{1}τ

11.5.1.2.2 Axioms of assertions

11.5.1.2.2.1 (x)(y)(z) x + (y + z) = (x + y) + z

11.5.1.2.2.2 (x) x + e = x

11.5.1.2.2.3 (x) x + x* = e

11.5.1.3 Evaluation of T3

11.5.1.3.1 T3 is consistent

11.5.1.3.2 T3 is not complete

11.5.1.3.3 T3 is independent

11.5.1.3.4 T3 is not categorical

11.5.2 T4 : Boolean algebras : first order predicate calculus with identity and operation symbols

11.5.2.1 Non-logical vocabulary

11.5.2.1.1 F^{1}, f^{2}, g^{2}, n, e

11.5.2.1.2 Convention

11.5.2.1.2.1 (τ U τ’) \(\Longleftarrow) f^{2}ττ’

11.5.2.1.2.2 (τ ∩ τ’) \(\Longleftarrow) g^{2}ττ’

11.5.2.1.2.3 Τ* \(\Longleftarrow) f^{1}τ

11.5.2.1.2.4 0 \(\Longleftarrow) n

11.5.2.1.2.5 1 \(\Longleftarrow) e

11.5.2.2 Axioms for assertions

11.5.2.2.1 (x)(y) x ∩ y = y ∩ x

11.5.2.2.2 (x)(y) x U y = y U x

11.5.2.2.3 (x)(y)(z) x ∩ ( y ∩ z ) = (x ∩ y) ∩ z

11.5.2.2.4 (x)(y)(z) x U ( y U z ) = (x U y) U z

11.5.2.2.5 (x)(y) x ∩ (x U y ) = x

11.5.2.2.6 (x)(y) x U (x ∩ y ) = x

11.5.2.2.7 (x)(y)(z) x ∩ ( y U z ) = (x ∩ y) U (x ∩ z)

11.5.2.2.8 (x)(y)(z) x U ( y ∩ z ) = (x U y) ∩ (x U z)

11.5.2.2.9 (x) x ∩ x* = 0

11.5.2.2.10 (x) x U x* = 1

11.5.2.2.11 0 != 1

11.5.2.3 Evaluation of T4

11.5.2.3.1 T4 is consistent

11.5.2.3.2 T4 is not complete

11.5.2.4 Dual of a theorem is a theorem

11.5.2.4.1 Dual : interchange of ∩ and U in a sentence of T4

11.6 Definition

11.6.1 To introduce notation that does not belong to the vocabulary of the theory but may improve readability of the formulas and render their content more clear

11.6.1.1 Metalinguistic sentence asserting that a given symbol of the object language means the same as some other object language expression

11.6.1.2 As an object language sentence of a certain form (usually an identity or biconditional or a generalized identity or biconditional, with the new symbol appearing on the left side alone or in a simple context)

11.6.2 Definition can lead to contradiction not just by its forms but also by the content of the theory

11.6.3 Criteria of Definition

11.6.3.1 Creative ( a definition ) : Generates new theorems in which the defined symbol does not occur

11.6.3.2 Definition should make the defined symbol eliminable

11.6.3.2.1 Resistance to circular definitions

11.6.3.3 Let T = (Δ,Γ) be theory. Let θ be a non-logical constant that does not belong to Δ. Let Φ be a sentence of L’ containing θ and otherwise formulated solely in terms of the vocabulary Δ. If we add Φ to the theses of T we get new theory T’ = (Δ U {θ} , {all sentences of T’ that are consequences of Γ U {Φ}}).

11.6.3.3.1 As a definition of θ related to T, Satisfies the criterion of eliminability ( a sentence Φ) : for every formula Ψ of T’, there is a formula χ of T’ s.t. all closures of Ψ\(\iff\)χ are theses of T’ and χ does not contain θ.

11.6.3.3.2 As a definition of θ related to T, Satisfies the criterion of non-creativity ( a sentence Φ) : every thesis of T’ that does not contain θ is a thesis of T

11.6.4 Construction of formally correct definitions

11.6.4.1 Let T be a theory formulated in L’, and suppose θ is a non-logical constant which does not occur in T.

11.6.4.2 If the constant θ is an n-ary predicate, Let definition be a sentence of form (α_{1})..(α_{n})(θα_{1}…α_{n} \(\iff\) ω), where ω is a formula of T and α_{1}…α_{n} are distinct and are all the variables occurring free in ω.

11.6.4.3 If the constant θ is a sentential letter, Let definition be a sentence θ \(\iff\) ω where ω is a sentence of T.

11.6.4.4 If the constant θ is an individual constant, Let definition be a sentence (α)(α = θ \(\iff\) ω) where ω is a formula of T and having α as its only free variable, and the corresponding sentence (∃β)(α)(α = β \(\iff\) ω) with β a variable distinct from α, is a thesis of T.

11.6.4.5 If the constant θ is an n-ary operation symbol,, Let definition be a sentence of form (γ_{1})..(γ_{n})(α)(α = θγ_{1}…γ_{n} \(\iff\) ω), where α, γ_{1}…α_{n} are distinct and are all the variables occurring free in ω. Θ does not occur in ω, and the corresponding uniqueness condition (γ_{1})..(γ_{n})(∃β)(α)(α = β\(\iff\) ω) with β distinct from α, γ_{1}, …, γ_{n} is a thesis of T.

11.6.5 Definability

11.6.5.1 Definable in the theory T ( a non-logical constant θ) : there is among the theses of T a formally correct definition of θ relative to the theory T – θ

11.6.5.1.1 T-θ : from the non-logical vocabulary of a theory T remove a constant θ, and from the theses of T we remove all those containing θ

11.6.5.2 Padoa’s principle : Non-logical constant θ is not definable in a theory T If we can give two models of T that differ in what they assign to θ but are otherwise the same.

Linear Algebra

Vector space 벡터 공간

(Section 2) 벡터공간

(subsection 2.1.) Vector Space

(표기법 2.1.1) \(F^{n} = \mathfrak{M}_{n,1}(F)\).

(정의 2.1.2) F가 위와 같고, 집합 V에 덧셈과 상수곱이 주어져 있을 때, 다음 연산규칙들을 만족하면 V를 F 위의 벡터공간 (vector space over F, 혹은 F-vector space) 라 부른다. 또, V의 원소들을 vector라 부르고, F의 원소를 scalar라 부른다.

\(u,v,w \in V\) 이면, \((u+v) +w = u + (v + w)\) (덧셈의 결합법칙)

\(v, w \in V\) 이면 , \(v + w = w + v\) (덧셈의 교환법칙)

(모든 \(v \in V\) 에 대하여 \( v + 0 = v\)) 인 \(0 \in V\) 존재 (덧셈의 항등원)

\(v \in V \)이면, \(v + (-v) = 0\) 인 \(-v \in V\) 존재. (덧셈의 역원)

\(a,b \in F\) ,\( v \in V\) 이면, \((a + b) v = av + bv\) (분배법칙)

\(a \in F\), \(v, w \in V\) 이면, \(a(v+w) = av + aw\) (분배법칙)

\(a, b \in F\), \(v \in V\) 이면, \(a(bv) = (ab)v\)

\(v \in V\) 이면, \(1v = v\)

(관찰 2.1.3)

vector space V 의 덧셈의 항등원 \(0 \in V\) 는 유일하다.

\(v \in V\) 이면, v의 덧셈의 역원 \(-v \in V\) 는 유일하다.

(표기법 2.1.4) 유일성이 보장된 \(0 \in W\) 를 the zero vector 라 부른다. \(v \in w\) 일 때, \(v -w = v + (-w)\) 로 쓴다.

(Cancellation law) (관찰 2.1.5) \(u,v,w \in V\) 이면, 다음 \( \left[u + v = u + w \right] \implies \left[ v = w \right] \) 이 성립한다. 즉, \(u + v = u + w\) 양 변에서 u 를 cancel 할 수 있다.

(관찰 2.1.7) \(v \in V\) 이고 \(a \in F\) 이면,

\(0v = 0\).

\(a0 = 0 \in V\).

\(-v = (-1) v\).

\(-(av) = (-a) v\).

\(-(av) = a(-v)\).

(subsection 2.2) Subspace

(정의 2.2.2) W가 F-vector space V 의 subset 일 때 (즉, \(W \subseteq V\) 일 때,) V로부터 물려받은 덧셈과 상수곱에 대하여 ,W 자신이 F-vector space 가 되면, 우리는 W를 V의 F-subspace (부분공간, 혹은 간단히 subspace) 라 부르고, \)W \le V\) 로 표기한다.

(관찰 2.2.3) F-vector space V 의 non-empty subset W 가 V의 subspace 일 필요충분조건은 다음과 같다. 즉, W가 덧셈과 상수곱에 대해 닫혀있으면 , V의 subspace 가 된다.

\(w_{1} , w_{2} \in W\) 이면, \(w_{1} + w_{2} \in W\).

\(w \in W\) 이고 \(a \in F\) 이면, \(aw \in W\).

(관찰 2.2.5) \(U \le W\) 이고 \(W \le V\) 이면, \(U \le V\) 이다. (subspace의 subspace는 subspace이다.)

(표기법 2.2.6) \(W_{1} , \cdots, W_{k} \subseteq U\) 일 때, \(\sum_{i = 1}^{k} W_{i} = W_{1} + \cdots + W_{k} = \lbrace w_{1} + \cdots + w_{k} | w_{1} \in W_{1}, \cdots, w_{k} \in W_{k} \rbrace \) 로 표기하고 \(W_{1} \cdots , W_{k}\) 의 합(sum)이라고 부른다.

(subspace 2.4) Isomorphism

(정의 2.4.1) V와 V’ 이 F-vector space 이고, 다음 조건

\(\phi(v_{1} + v_{2}) = \phi(v_{1}) + \phi(v_{2}) , (v_{1}, v_{2} \in V) \)

\(\phi(av) = a \phi(v) (v \in V, a \in F)\)

을 만족하는 bijection \(\phi: V \to V’\) 이 존재하면, 우리는 F-vector space V 와 V’ 이 isomorphic 하다 말하고, \(V \approx V’ \)혹은\begin{tikzpicture} \matrix (m) [matrix of math nodes,row sep=3em,column sep=4em,minimum width=2em] {V &V’};\path-stealth edge node [above] {\)\phi\(} node [below]{\)\approx\)} (m-1-2); \end{tikzpicture}로 표기한다. 이 때, $$\phi$$ 를 (F-vector space) isomorphism (from V onto V’) 이라고 부른다.(관찰 2.4.2) $$\phi : V \to W$$ 가 isomorphism 이면, $$\phi(0) = 0$$ 이고, 모든 $$v \in V$$에 대해 $$\phi(-v) = -\phi(v)$$ 이다.(관찰 2.4.4) 위 (정의 2.4.1) 의 relation $$\approx$$ 는 equivalence relation 이다.

Basis and Dimension 기저와 차원

(Section 3) 기저와 차원

(subsection 3.1) Linear Combination

(정의 3.3.1) (가)\( v_{1}, \cdots, v_{n} \in V\) 일 때, \(a_{1}v_{1} + a_{2}v_{2} + \cdots + a_{n}v_{n} = \sum_{i = 1}^{n} a_{i}v_{i}\) , (단, \(a_{1}, \cdots, a_{n} \in F\)) 꼴의 vector를 우리는 \({v_{1}, \cdots, v_{n}}\) 의 일차결합 (F-linear combination) 이라고 부른다.

(나) \(S \subset V\) 일 때, (S가 무한집합이라면)\(v \in V\) 가 S의 일차결합이라는 말은 v가 [\(S_{0}\) 의 일차결합] 인 유한집합 \(S_{0} \subseteq S\) 가 존재하냐는 뜻이다. (매번 \(S \neq 0\) 이라는 조건을 다느니,\(\0 \) 의 일차결합 전체의 집합은 {0} 이라고 약속하는 것이 간편하다.)

(관찰 3.1.3) \(S \subseteq V\) 일 때, [S 의 linear combination 전체의 집합] 은 V의 부분공간이 된다.

(정의 3.1.4) \(S \subseteq V\) 일 때, S를 포함하는 가장 작은 V의 subspace를 \(\langle S \rangle\) 로 표기하고,\( \langle S \rangle\) 를 subspace generated by S (혹은 subspace spanned by S, S 가 생성한 부분공간) 이라고 부른다. (여기에서, [S를 포함하는 가장 작은 V의 subspace] 라는 말은 \(S \subseteq \langle S \rangle \le V\) 이면서, 만약 \(S \subseteq W \le V\) 이면, \(\langle S \rangle \le W\) 라는 뜻이다.)

(보조정리 3.1.5)\( S \subseteq V\) 이면, \(\langle S \rangle = \bigcap_{S \subseteq W \le V} W\) 이다. 이로부터 \(\langle S \rangle\) 의 existence 와 uniqueness 가 얻어진다.

(명제 3.1.6) \(S \subseteq V\) 이면,\( \langle S \rangle\) 는 [S 의 linear combination 전체의 집합] 과 같다.

(subsection 3.2) 일차독립과 일차종속

(정의 3.2.1) (가) 유한집합 \(\lbrace v_{1}, \cdots, v_{n}\rbrace \subseteq V\) 가 다음 조건 \(a_{1}v_{1} + a_{2}v_{2} + \cdots + a_{n}v_{n} = 0\) 이면, \(a_{1} = \cdots = a_{n} = 0 (단, a_{i} \in F )\) 를 만족하면, 우리는 \({v_{1}, \cdots, v_{n}}\) 을 일차독립 (F-linearly independent) 인 부분집합이라고 부른다.

(나) \(\0 \neq S \subseteq V\) 일 때, S가 일차독립이라는 말은 S의 모든 finite subset 이 일차독립이라는 뜻이다.

(다) \(\0 \neq S \subseteq V\) 가 일차독립이 아니면 일차종속 (F-linearly dependent) 이라고 말한다.

(관찰 3.2.2) 유한집합 S 가 일차독립이면, S의 모든 non-empty subset 도 일차독립이다.

(관찰 3.2.7) \({v_{1}, \cdots, v_{n}} \subseteq V\) 일 때, 다음 세 조건은 동치이다.

\({v_{1}, \cdots, v_{n}}\) 는 일차독립이다.

\({v_{1}, \cdots, v_{n}}\) 의 일차결합으로 zero vector 0 를 표현하는 방법은 하나 – 즉 \(0v_{1} + \cdots, 0v_{n} = 0 –\) 뿐이다.

어떤\( v \in V\) 가 \({v_{1}, …, v_{n}}\) 의 일차결합으로 표현된다면, 그 표현법은 하나뿐이다.

(subsection 3.3) Vector space 의 Basis

(정의 3.3.1) V가 vector space 이고 \(\0 \neq \mathfrak{B} \subseteq V\) 일 때,

\(\langle \mathfrak{B} \rangle = V\) (즉 \(\mathfrak{B}\) generates V) ,

\(\mathfrak{B}\) 는 linearly independent

이면, 우리는\( \mathfrak{B}\) 를 V의 기저 (basis, F-basis) 라고 부른다.

(관찰 3.3.2) 다음 조건은 동치이다.

\(\mathfrak{B}\) 는 V의 basis 이다.

V의 모든 vector 는 \(\mathfrak{B}\) 의 linear combination 으로 표현할 수 있고, 그 표현법은 하나 뿐이다.

(정의) \(e_{i} \in F^{n}\) 을 i-번째 좌표만 1 이고 나머지 좌표는 0인 표준단위벡터라고 하면, \(\mathcal{E} = {e_{1}, …, e_{n}}\) 은 \(F^{n}\) 의 기저이다. 우리는 \(\mathcal{E}\) 를 \(F^{n}\) 의 표준기저 (standard basis, Euclidean basis) 라고 부른다.

(정의 3.3.5) 유한집합 \(\mathfrak{B} = {v_{1}, …, v_{n}}\) 이 V의 F-basis 라고 하자. \(v \in V\) 가 \(v = a_{1}v_{1} + a_{2}v_{2} + \cdots + a_{n}v_{n}\) 으로 표현될 때 (이러한 표현은 항상 유일한 방법으로 가능하다) 우리는 \(F^{n}\) 의 vector \((a_{1}, …, a_{n})\) 을 [basis \(\mathfrak{B}\) 에 관한 v의 좌표] 라고 부르고, \(\left[v\right]{\mathfrak{B}} = ^{t}(a{1}, …, a_{n})\) 으로 표기한다.

(관습 3.3.6) \(v_{1}, …, v_{n}\) 의 순서가 고정된 기저 \(\mathfrak{B} = \lbrace_{1}, …, v_{n}\rbrace\) 을 ordered basis 라고 부른다. 더불어 우리가 단순히 basis라 말할 때도 언제나 ordered basis 를 의미하는 것으로 약속한다. 즉, basis = ordered basis 가 우리의 관습이다.

(명제 3.3.11)\( A = (a_{ij}) \in \mathfrak{M}{n,n} (F)\) 가 가역이고, \(\lbrace v{1}, …, v_{n}\rbrace\) 이 V의 기저일 때, \(w_{j} = \sum_{i = 1}^{n} a_{ij} v_{i} (j = 1,…,n) \)이라고 정의하면, \({w_{1}, …, w_{n}}\) 도 V의 기저이다.

(따름명제 3.3.12)\( A = (a_{ij}) \in \mathfrak{M}_{n,n} (F)\) 가 가역이면, \(\lbrace \left[ A \right]^{1}, …., \left[ A \right]^{n} \rbrace\) 은 \(F^{n}\) 의 기저이다. (단, \(\left[ A \right] ^{j}\) 는 행렬 A의 j-th column.)

(subsection 3.4) Basis의 존재

(정리 3.4.3) 모든 non-zero vector space 는 basis 를 갖는다.

(section 3.5) Vector space 의 Dimension

(정리 3.5.1) Vector space V 가 basis \(\mathfrak{B}\) 와 \(\mathfrak{C}\) 를 가지면, \(|\mathfrak{B}| = |\mathfrak{C}|\) 이다.

(보조정리 3.5.2) Vector space V 가 finite basis \(\mathfrak{B} = {v_{1}, …, v_{n}}\) 를 갖는다고 하자. 이 때, 만약\( \mathfrak{C} = {w_{1}, …, w_{n}} \subseteq V\) 이고 n < m 이면, \(\mathfrak{C}\) 는 linearly independent 이다.

(정의 3.5.4) 벡터공간 V가 F-basis \(\mathfrak{B}\) 를 가질 때, \(\mathfrak{B}\) 의 원소수 \(|\mathfrak{B}|\) 를 V의 차원(dimension) 이라 부르고, \(dim_{F} V = dim V\) 로 표기한다. (dim 0 = 0) dim V 가 유한이면, 우리는 V를 유한차원 (finite dimensional) 벡터공간이라고 부른다. 무한이면 무한차원 벡터공간이라 부른다.

finite dimensional vector space 를 앞으로 f.d.v.s 라 줄여서 표기한다.

(Basis extension theorem) (정리 3.5.5) S가 V의 linearly independent subset이면, S를 포함하는 V의 basis 가 존재한다.

(Basis extension theorem) (따름정리 3.5.6) V가 f.d.v.s. 이고 \(W \le V\) 라고 하자. 만약 \({w_{1}, …, w_{r}}\) 이 W의 기저이면 이를 확장하여 V의 기저 \({w_{1}, …, w_{r}, v_{1}, …, v_{s}}\) 를 찾을 수 있다. (단, \(s \ge 0\))

(따름정리 3.5.8) V가 f.d.v.s. 이고 \(W \le V\) 이면,

W도 f.d.v.s. 이고, \(dim W \le dim V\) 이다.

만약 \(dim W = dim V\) 이면, \(W = V\) 이다.

(따름정리 3.5.9) \(S \subseteq V\) 이고 \(|S| = dim V < \infty\) 이면, 다음 조건은 동치이다.

S는 V의 기저이다.

S는 일차독립이다.

\(\langle S \rangle = V\) 이다.

(subsection 3.6) 우리의 철학

(관찰 3.6.6) \(\phi : V \to W\) 가 isomorphism 일 때, \(\mathfrak{B}\) 가 V의 기저이면, \(\phi( \mathfrak{B}\) 는 W의 기저이고, 따라서 \(dim V = dim W\) 이다.

(우리의 철학 1) Isomorphism 의 철학

up to isomorphism 같은 벡터공간은 덧셈과 상수곱에 의해 묘사되는 성질이 같다.

(우리의 철학 2) Identification 의 철학

up to isomorphism 같고 표기법만 다르다면 표기법을 고쳐서 표기법도 같게 만들면 된다.

Linear Mapping 선형 사상

(Section 4) 선형사상

(subsection 4.1) Linear map

(정의 4.1.1) V,W 가 F-위의 벡터공간일 때, 함수 \(L : V \to W\) 가 다음 조건을 만족하면 L을 linear map (선형사상, linear mapping, linear transformation from V into W) 라고 부른다.

\(L(v_{1} + v_{2}) = L(v_{1}) + L(v_{2}) (v_{1}, v_{2} \in V)\)

\(L(av) = aL(v) ( v \in V , a \in F)\)

(관찰 4.1.2) \(L : V \to W\) 가 linear map 일 때,

\( L(0) = 0\)

\(v \in V\) 이면, \(L(-v) = -L(v)\)

\(u,v \in V\) 이면, \(L(u-v) = L(u) – L(v)\)

(정의 4.1.3) \(L : V \to W\) 가 linear map일 때,

L 이 injective (단사) 이면, L을 monomorphism 이라 부른다.

L 이 surjective (전사) 이면, L 을 epimorphism 이라고 부른다.

L 이 bijective (전단사) 이면, L을 isomorphism 이라고 부른다.

V = W 이면, L 을 endomorphism, 혹은 Linear operator, 혹은 간단히 operator 라 부른다.

Bijective endomorphism 은 automorphism 이라고 부른다.

(관찰 4.1.4) \(L : V \to W\) 가 linear map 일 때, 다음 조건은 동치이다.

L 은 isomorphism.

[\(M \bullet L = I_{V}\) 이고\( L \bullet M = I_{W}\) ] 인 linear map\(M : W \to V\) 가 존재.

(정의 4.1.5)\( L : V \to W\) 가 linear map 일 때,

\(ker L = L^{-1} (0) = {v \in V | L(v) = 0}\) 을 L 의 kernel 이라고 부른다.

\(im L = L(V) = {L(v) | v \in V}\) 를 L의 image 라고 부른다.

(관찰 4.1.6) \(L : V \to W\) 가 linear map 이면, \(ker L \le V , im L \le W\) 이다.

(관찰 4.1.8) \(L : V \to W\) 가 linear map 일 때, 다음 조건은 동치이다.

L 은 monomorphism 이다.

\(u,v, \in V\) 이고 \(Lu = Lv\) 이면, \(u = v \)이다.

\(v \in V\) 이고\(Lv = L0\) 이면, \(v = 0\) 이다.

\(ker L = 0\) 이다.

(관찰 4.1.9)\( L : V \to W\) 가 linear 이고, \(S \subseteq V\) 이면,\( L\langle S \rangle = \langle LS \rangle\) 이다.

(관찰 4.1.12) \(L : V \to W\) 가 linear map 일 때, 다음 조건은 동치이다.

L 은 isomorphism 이다.

L 은 basis 를 basis 로 옮긴다.

(표기법 4.1.13) \(U \le V\) 이고 \(v \in V\) 일 때, \(v + U = {v + u | u \in U}\) 의 notation 을 쓴다.

(subsection 4.2) Linear map 의 보기

(관찰 4.2.7) 두 선형사상의 합성은 다시 선형사상이다. 즉, \(M : U \to V\) 와 \(L : V \to W\) 가 linear 이면, \(L \bullet M : U \to W\) 도 linear 이다.

(정의 4.2.10) \(A \in \mathfrak{M}{m,n} (F)\) 일 때, \(L{A} : F^{n} \to F^{m}\) 을 \(L_{A}(X) = AX (X \in F^{n})\) 으로 정의하면, \(L_{A}\) 가 linear map 인 것은 자명하다. \(L_{A}\) 를 matrix A 에 대응하는 linear map 이라 부른다.

(subsection 4.3) Dimension Theorem

(Dimension Theorem) (정리 4.3.1) V, W 가 f.d.v.s 이고, \(L : V \to W\) 가 linear map 이면, dim V = dim ker L + dim im L 이다.

(따름정리 4.3.2) V,W 가 f.d.v.s. 이고 \(dim V = dim W\) 일 때, \(L : V \to W\) 가 linear map 이면 다음 세 조건은 동치이다.

L 은 isomorphism. (즉, L은 bijection)

L 은 monomorphism (즉, L 은 injection)

L 은 epimorphism (즉, L 은 surjection)

(Pigeonhole Principle) (정리 4.3.4) X,Y 가 (non-empty) finite set 이고 \(|X| = |Y|\) 일 때, \(f : X \to Y\) 가 함수이면 다음 세 조건은 동치이다.

f는 bijection.

f 는 injection.

f 는 surjection.

(따름정리 4.3.6) V,W 가 f.d.v.s. 이고 \(L : V \to W\) 가 linear map 일 때,

L 이 monomorphism 이면, \(dim V \le dim W\).

L 이 epimorphism 이면, \(dim V \ge dim W\).

(subsection 4.4) Rank Theorem

(정의 4.4.1) \(A \in \mathfrak{M}{m,n} (F)\) 일 때, A 의 i-번째 row 는 \(\left[ A \right]{i}\) 로 표기하고, A의 j-번째 column 은 \(\left[ A \right]^{j}\) 로 표기한다. \(\mathfrak{M}{1,n}(F)\) 의 부분공간 \(\langle \left[ A\right]{1}, …, \left[ A \right]_{m} \rangle\) 을 A의 rowspace 라 부른다. A 의 row space 의 dimension 은 A 의 row rank 라 부른다. 마찬가지로, A의 column space 는 \(F^{m}\) 의 부분공간 \(\langle \left[ A \right]^{1} , …, \left[ A \right]^{n} \rangle\) 이고, 그 dimension 은 A의 column rank 라고 부른다.

(관찰 4.4.3) 행렬 A에 elementary row operation 을 수행해도 A의 row space 는 변화하지 않는다. 따라서, [A 로부터 얻어지는 row-reduced echelon form 의 row space] 와 [A 의 row space ] 는 같다.

(Rank Theorem) (정리 4.4.4) \(A \in \mathfrak{M}_{m,n} (F)\) 이면, [row rank of A] = [column rank of A] 이다.

(정의 4.4.5) \(A \in \mathfrak{M}_{m,n}(F)\) 일 때, A의 row rank (혹은 column rank) 를 간단히 A의 rank 라 부르고 rk(A) 로 표기한다.

(따름정리 4.4.6) \(A \in \mathfrak{M}_{m,n} (F)\) 일 때, homogeneous linear equation (*) AX = 0 의 solution space 의 dimension 은 n – rk(A) 이다.

(따름정리 4.4.7) 행렬 A에 elementary row operation 을 수행해도 (A의 column space 는 변화하지만) A의 column rank 는 변화하지 않는다.

(subsection 4.5) Linear extension theorem

(관찰 4.5.1) \(\mathfrak{B} = {v_{1}, …, v_{n}}\) 이 V의 basis 이고 \(L, M : V \to W\) 가 linear 일 때, [\(L(v_{i}) = M(v_{i})\) for all i] 이면, \(L = M\) 이다. 즉, 선형사상은 기저에서의 값이 결정한다.

(Linear extension theorem) (정리 4.5.3) \(\mathfrak{B}\) 가 V의 basis 이고\( f : \mathfrak{B} \to W\) 가 함수이면, \(L|_{\mathfrak{B}} = f\) 인 linear map\( L : V \to W\) 가 유일하게 존재한다.

(Classification of Vector Spaces) (따름정리 4.5.10) V,W 가 F-vector space 일 때, 다음은 동치이다.

\(V \approx W\)

\(dim V = dim W\)

(Classification of f.d.v.s) (따름정리 4.5.11) f.d.v.s. V 의 dimension 이 n 이면, \(V \approx F^{n}\) 이다.

Fundamental Theorem 기본정리

(Section 5) 기본정리

(subsection 5.1) Vector space of linear maps

(정의 5.1.1) V,W 가 F-vector space 일 때, \(\mathfrak{L} (V,W) = {L : V \to W | L 은 linear map}\( 으로 정의한다. 그리고, [\(L,M \in \mathfrak{L}(V,W)\) 의 합\( L + M\) ] 과, [scalar \(a \in F\) 와\(L \in \mathfrak{L}(V,W)\) 의 상수곱 \(aL\)] 을 각각 \((L + M) (v) = L(v) + M(v) , (aL)(v) = aL(v) (v \in V)\) 로 정의하면 \(\mathfrak{L} (V,W)\) 도 벡터공간이 된다.

(정의 5.1.5) V가 벡터공간일 때, \(V^{} = \mathfrak{L} (V,F)\) 로 정의하고, 우리는 \(V^{}\) 를 V 의 dual space (쌍대공간) 이라고 부른다. 또, \(V^{}\) 의 원소는 linear functional 이라고 부른다. 뿐만 아니라, 우리는 dual 의 dual , 즉 \((V^{})^{} = V^{**} = \mathfrak{L}(V^{} , F)\) 도 생각하게 된다. \(V^{**}\) 는 V의 double dual 이라고 부른다.

(관찰 5.1.7) V가 f.d.v.s. 이면, \(dim V^{*} = dim V\) 이다.

(정의 5.1.8) f.d.v.s V 에 대해 우선\( \mathfrak{B} = {v_{1}, …, v_{n}}\) 을 V의 기저라고 하자. \(v_{i}^{} \in V^{} 를 v_{i}^{} (v_{j}) = \delta_{ij} (1 \le i,j \le n )\) 이라 정의한다. 이제 \(\mathfrak{B}^{} = {v_{1}^{} , …, v_{n}^{}}\) 가 \(V^{}\) 의 basis 이다. 이 \(\mathfrak{B}^{} = {v_{1}^{} , …, v_{n}^{}}\) 를 \(\mathfrak{B}\) 의 dual basis 라 부른다.

(관찰 5.1.9) V,W 가 f.d.v.s. 이면, \(dim \mathfrak{L} (V,W) = (dim V) \cdot (dim W)\) 이다.

(subsection 5.2.) 기본정리 ; 표준기저의 경우

(표기법 5.2.1) 이 절에서는 다음과 같은 notation 을 사용한다.

\(V = F^{n}\) ; \(dim V = n\) , [\(F^{n}\) 의 표준기저 ] = \(\mathcal{E} = {e_{1}, …, e_{n}}\) .

\(W = F^{m}\) ; \(dim W = m\) , [\(F^{m}\) 의 표준기저 ] = \(\mathcal{F} = {f_{1}, …, f_{m}}\) .

\(U = F^{r}\) ; \(dim U = r\) , [\(F^{r}\) 의 표준기저 ] = \(\mathcal{G} = {g_{1}, …, g_{r}}\) .

(정의 5.2.2)\(L \in \mathfrak{L}(F^{n} , F^{m})\) 일 때,

\(L(e_{1}) = \begin{pmatrix}a_{11}

a_{21}

\vdots

a_{m1} \end{pmatrix} , L(e_{2}) = \begin{pmatrix}a_{12}

a_{22}

\vdots

a_{m2} \end{pmatrix} , \cdots , L(e_{n}) = \begin{pmatrix}a_{1n}

a_{2n}

\vdots

a_{mn} \end{pmatrix} \)

이라고 표기하자. (단,\( a_{ij} \in F\)) 이를 한 줄로 줄이면 \(L(e_{j}) = \sum_{i = 1}^{j} a_{ij} \mathbf{f}_{i} ( j = 1,…,n)\) 이 된다. 이때, 우리는

\(\left[L\right]{\mathcal{F}}^{\mathcal{E}} = \left[L\right] = (a{ij}) = \begin{pmatrix} a_{11} & a_{12} & a_{13} & \cdots & a_{1n}

a_{21} & a_{22} & a_{23} & \cdots & a_{2n}

\vdots & \vdots & \vdots & \ddots & \vdots

a_{m1} & a_{m2} & a_{m3} & \cdots & a_{mn} \end{pmatrix}\)

으로 정의하고, \(\left[L\right]_{\mathcal{F}}^{\mathcal{E}} = \left[L\right]\) 를 표준기저 \(\mathcal{E}\) 와 \(\mathcal{F}\) 에 관한 L 의 행렬(행렬표현) 이라고 부른다. matrix of L with respect to \(\mathcal{E}\) and \(\mathcal{F}\) 라고 부른다.

(선형대수학의 기본정리) (정리 5.2.3) 함수 \(\Phi_{\mathcal{F}}^{\mathcal{E}} = \Phi : \mathfrak{M}{m,n} (F) \to \mathfrak{L}(F^{n} , F^{m})\) 을 \(\Phi(A) = L{A} (A \in \mathfrak{M}{m,n} (F))\) 으로 정의하자. 또, 함수 \(\Psi{\mathcal{F}}^{\mathcal{E}} = \Psi : \mathfrak{L} (F^{n}, F^{m}) \to \mathfrak{M}{m,n} (F)\) 를 \(\Psi(L) = \left[L\right]{\mathcal{F}}^{\mathcal{E}} = \left[L\right], (L \in \mathfrak{L}(F^{n}, F^{m}))\) 이라고 정의하면,

\(\Phi\) 는 isomorphism 이고 \(\Psi\) 는 그의 inverse map 이다.

\(B \in \mathfrak{M}{r,m} (F)\) 이고 \(A \in \mathfrak{M}{m,n}(F)\) 이면, \(\Phi_{\mathcal{G}}^{\mathcal{F}} (B) \bullet \Phi_{\mathcal{F}}^{\mathcal{E}} (A) = \Phi_{\mathcal{G}}^{\mathcal{E}} (BA)\) , 즉 \(L_{B} \bullet L_{A} = L_{BA}\) 가 성립한다. 또, \(L \in \mathfrak{L}(F^{n}, F^{m})\) 이고 \(M \in \mathfrak{L}(F^{m} , F^{r})\) 이면, \(\Psi_{\mathcal{G}}^{\mathcal{F}} (M) \cdot \Psi_{\mathcal{F}}^{\mathcal{E}} (L) = \Psi_{\mathcal{G}}^{\mathcal{E}} (M \bullet L)\) , 즉 \(\left[M\right]_{\mathcal{G}}^{\mathcal{F}} \cdot \left[L\right] {\mathcal{F}}^{\mathcal{E}} = \left[M \bullet L\right]{\mathcal{G}}^{\mathcal{E}}\) 가 성립한다.

(따름정리 5.2.5)

모든 linear map \(L : F^{n} \to F^{m}\) 은 \(L_{A}\) 의 꼴이고, 이 때, \(A \in \mathfrak{M}_{m,n} (F)\) 는 유일하게 결정된다.

\(A \in \mathfrak{M}{m,n}(F)\) 이면, \(\left[L{A}\right] = A\) 이다.

\(L \in \mathfrak{L}(F^{n}, F^{m})\) 이면, \(L_{\left[L\right]} = L\) 이다. 따라서, \(L(X) = \left[L\right] \cdot X , (X \in F^{n})\) 이다.

(subsection 5.3) 기본정리 ; 일반적인 경우

(표기법 5.3.1) 이 절에서는 다음과 같은 notation 을 사용한다.

V; \(dim V = n\) , [V 의 기저 ] = \(\mathfrak{B} = {v_{1}, …, v_{n}}\) .

W; \(dim W = m\) , [W 의 기저 ] = \(\mathfrak{C} = {w_{1}, …, w_{m}}\) .

U; \(dim U = r\) , [U 의 기저 ] = \(\mathfrak{D} = {u_{1}, …, u_{r}}\) .

위의 기저들은 모두 ordered basis 이며 모두 고정된 (fixed) 것으로 생각한다.

(정의 5.3.2) \(L \in \mathfrak{L} (V,W)\) 일 때

\(\begin{cases} L(v_{1}) = a_{11}w_{1} + a_{21}w_{2} + \cdots + a_{m1}w_{m}

L(v_{2}) = a_{12}w_{1} + a_{22}w_{2} + \cdots + a_{m2}w_{m}

\vdots

L(v_{n}) = a_{1n}w_{1} + a_{2n}w_{2} + \cdots + a_{mn}w_{m} \end{cases} \)

이라고 표기하자. (단,\( a_{ij} \in F\)). 이를 한 줄로 줄이면 \(L(v_{j}) = \sum_{i = 1}^{m} a_{ij} w_{i} (j = 1, …, n)\) 이 된다. 이 때, 우리는

\(\left[L\right]{\mathfrak{C}}^{\mathfrak{B}} = (a{ij}) = \begin{pmatrix} a_{11} & a_{12} & a_{13} & \cdots & a_{1n}

a_{21} & a_{22} & a_{23} & \cdots & a_{2n}

\vdots & \vdots & \vdots & \ddots & \vdots

a_{m1} & a_{m2} & a_{m3} & \cdots & a_{mn} \end{pmatrix}\)

으로 정의하고, \(\left[L\right]{\mathfrak{C}}^{\mathfrak{B}} \in \mathfrak{M}{m,n}(F)\) 를 기저 \(\mathfrak{C}\) 와 \(\mathfrak{B}\) 에 관한 L 의 행렬(행렬표현) 이라고 부른다. matrix of L with respect to \(\mathfrak{C}\) and \(\mathfrak{B}\) 라고 부른다.

(선형대수학의 기본정리) (정리 5.3.5) 함수 \(\Phi_{\mathfrak{C}}^{\mathfrak{B}} : \mathfrak{M}{m,n} (F) \to \mathfrak{L}( V, W )\) 을 \(\left[(\Phi{\mathfrak{C}}^{\mathfrak{B}} (A))(v)\right] {\mathfrak{C}} = A \cdot \left[v\right]{\mathfrak{B}} (A \in \mathfrak{M}{m,n} (F) , v \in V)\) 으로 정의하자. 또, 함수 \(\Psi{\mathfrak{C}}^{\mathfrak{B}} : \mathfrak{L} (V , W ) \to \mathfrak{M}{m,n} (F)\) 를 \(\Psi{\mathfrak{C}}^{\mathfrak{B}} (L) = \left[L\right] _{\mathfrak{C}}^{\mathfrak{B}} (L \in \mathfrak{L}(F^{n}, F^{m}))\) 이라고 정의하면,

\(\Phi_{\mathfrak{C}}^{\mathfrak{B}}\) 는 isomorphism 이고 \(\Psi_{\mathfrak{C}}^{\mathfrak{B}}\) 는 그의 inverse map 이다.

\(B \in \mathfrak{M}{r,m} (F)\) 이고 \(A \in \mathfrak{M}{m,n}(F)\) 이면,\( \Phi_{\mathfrak{D}}^{\mathfrak{C}} (B) \bullet \Phi_{\mathfrak{C}}^{\mathfrak{B}} (A) = \Phi_{\mathfrak{D}}^{\mathfrak{B}} (BA)\) 가 성립한다. 또,\( L \in \mathfrak{L}(V, W )\) 이고 \(M \in \mathfrak{L}(W , U )\) 이면, \(\Psi_{\mathfrak{D}}^{\mathfrak{C}} (M) \cdot \Psi_{\mathfrak{C}}^{\mathfrak{B}} (L) = \Psi_{\mathfrak{D}}^{\mathfrak{B}} (M \bullet L)\) , 즉 \(\left[M\right]{\mathfrak{D}}^{\mathfrak{C}} \cdot \left[L\right]{\mathfrak{C}}^{\mathfrak{B}} = \left[M \bullet L\right] _{\mathfrak{D}}^{\mathfrak{B}}\) 가 성립한다.

(따름정리 5.3.6) \(L \in \mathfrak{L} (V,W)\) 이고 \(v \in V\) 이면, \(\left[L(v)\right]{\mathfrak{C}} = \left[L\right]{\mathfrak{C}}^{\mathfrak{B}} \cdot \left[v\right]_{\mathfrak{B}}\) 이다.

(subsection 5.4) 기본정리의 결과와 우리의 철학

(우리의 철학 3) 행렬과 선형사상은 같은 것이다. (단, [곱셈] = [합성])

(따름정리 5.4.12) \(A \in \mathfrak{M}_{n,n}(F)\) 일 때, 다음 조건은 동치이다.

\(A\)는 invertible matrix.

\(L_{A}\) :\( F^{n} \to F^{n}\) 은 isomorphism.

이 때, \((L_{A})^{-1} = L_{A^{-1}}\) 이다.

(총정리 5.4.17) \(A \in \mathfrak{M}_{n,n}(F)\) 일 때, 다음 조건들은 동치이다.

\(A\)는 invertible matrix.

\(A\) 는 left inverse 를 갖는다. 즉 \(BA = I \)인 \(B \in \mathfrak{M}_{n,n}(F)\) 가 존재.

\(A\) 는 right inverse 를 갖는다. 즉 \(AB = I\) 인 \(B \in \mathfrak{M}_{n,n}(F)\) 가 존재.

\(L_{A} : F^{n} \to F^{n}\) 은 isomorphism

\(L_{A} : F^{n} \to F^{n}\) 은 monomorphism

\(L_{A} : F^{n} \to F^{n}\) 은 epimorphism

\({\left[A\right]^{1} , …, \left[A\right]^{n}}\) 은 \(F^{n}\) 의 basis.

\({\left[A\right]^{1}, …, \left[A\right]^{n}}\) 은 일차독립

\(\langle \left[A\right]^{1} , …, \left[A\right]^{n} \rangle = F^{n}\) .

\(rk(A) = n.\)

\(AX = B\) 는 unique solution 을 갖는다.

\(AX = 0\) 은 trivial solution 만을 갖는다.

위 조건들에 \(A\) 대신 \(^{t}A\) 를 대입한 모든 조건들.

(우리의 철학 4) 선형사상 \(L : V \to W\) 는\( L_{A} : F^{n} \to F^{m}\) 과 같은 함수이다.

(subsection 5.5) Change of Bases

(표기법 5.5.1) 이 절에서는

V; \(dim V = n\) , [V 의 기저 ] = \(\mathfrak{B} = {v_{1}, …, v_{n}}\), \(\mathfrak{B}’ = {v’{1}, …, v’{n}}\)

W; \(dim W = m\) , [W 의 기저 ] = \(\mathfrak{C} = {w_{1}, …, w_{m}}\) , \(\mathfrak{C} = {w’{1}, …, w’{m}}\)

U; \(dim U = r\) , [U 의 기저 ] = \(\mathfrak{D} = {u_{1}, …, u_{r}}\) .

(따름정리 5.5.2) \(L \in \mathfrak{L} (V,W)\) 이면, \(\left[L\right]{\mathfrak{C}’}^{\mathfrak{B}’} = \left[I{W}\right]{\mathfrak{C}’}^{\mathfrak{C}} \cdot \left[L\right]{\mathfrak{C}}^{\mathfrak{B}} \cdot \left[I_{W}\right]_{\mathfrak{B}}^{\mathfrak{B}’}\)

(따름정리 5.5.3) 다음이 성립한다.

\(\left[L\right]{\mathfrak{C}}^{\mathfrak{B’}} = \left[L\right]{\mathfrak{C}}^{\mathfrak{B}} \cdot \left[I\right]_{\mathfrak{B}}^{\mathfrak{B}’}\)

\(\left[L\right]{\mathfrak{C}’}^{\mathfrak{B}} = \left[I\right]{\mathfrak{C}’}^{\mathfrak{C}} \cdot \left[L\right]_{\mathfrak{C}}^{\mathfrak{B}}\)

(정의 5.5.4) V의 기저 \(\mathfrak{B}, \mathfrak{B}’\) 에 대해, \(\left[I\right]{\mathfrak{B}}^{\mathfrak{B}’}\) 혹은 \(\left[I\right]{\mathfrak{B}’}^{\mathfrak{B}}\) 을 transition matrix (한자 행렬) 라 부른다. Transition matrix 는 기저 변환 (change of bases) 의 정보를 갖고 있다.

(관찰 5.5.5) \(\left[I\right]{\mathfrak{B}}^{\mathfrak{B}’} \cdots \left[I\right]{\mathfrak{B}’}^{\mathfrak{B}} = I\) , 즉 transition matrix 는 가역이고 \((\left[I\right]{\mathfrak{B}’}^{\mathfrak{B}})^{-1} = \left[I\right]{\mathfrak{B}}^{\mathfrak{B}’}\). 역으로, 가역행렬은 항상 transition matrix 로 인식할 수 있다.

(관찰 5.5.6) \(U \in \mathfrak{M}_{n,n} (F)\) 가 가역행렬이고 \(\mathfrak{B}\) 가 V의 기저이면, 다음이 성립한다.

\(U = \left[I\right]_{\mathfrak{B}}^{\mathfrak{B}’}\) 인 V 의 기저 \(\mathfrak{B}’\) 가 존재한다.

따라서, \(U = \left[I\right]_{\mathfrak{B}’’}^{\mathfrak{B}}\) 인 V 의 기저 \(\mathfrak{B}’’\) 도 존재한다.

(관찰 5.5.7) \(\mathfrak{B}, \mathfrak{C}\) 가 각각 \(F^{n}, F^{m}\) 의 basis 이고, \(A \in \mathfrak{M}{m,n}(F)\) 이면, 기본정리에 의해 \(\left[L{A}\right]{\mathcal{F}}^{\mathcal{E}} = A\) 이므로, \(\left[L{A}\right]{\mathfrak{C}}^{\mathfrak{B}} = \left[I\right]{\mathfrak{C}}^{\mathcal{F}} \cdot A \cdot \left[I\right]_{\mathcal{E}}^{\mathfrak{B}}\) 가 성립한다.

(명제 5.5.8) \(A,B \in \mathfrak{M}_{m,n} (F)\) 일 때 다음은 동치이다.

\(QAP = B\) 인 가역행렬 \(Q \in \mathfrak{M}{m,m}(F)\) 와 가역행렬 \(P \in \mathfrak{M}{n,n}(F)\) 가 존재한다.

\(\left[L_{A}\right]_{\mathfrak{C}}^{\mathfrak{B}} = B\) 인\( F^{n}\) 의 basis \(\mathfrak{B}\) 와\( F^{m}\) 의 basis \)\mathfrak{C}\( 가 존재한다.

다음 diagram

%\begin{tikzpicture} \matrix (m) [matrix of math nodes,row sep=3em,column sep=4em,minimum width=2em] { F^{n} & F^{m}

F^{n} & F^{m}

}; \path[-stealth] (m-1-1) edge node [left] {\(\alpha_{\mathcal{E}}^{\mathfrak{B}} \)} node [right] {\(\approx\)} (m-2-1) edge node [above] {\(L_{A}\)} (m-1-2) (m-1-2) edge node [right] {\(\alpha_{\mathcal{F}}^{\mathfrak{C}} \)} node [left] {\(\approx\)} (m-2-2) (m-2-1) edge node [below] {\(L_{B}\)} (m-2-2);\end{tikzpicture}

이 commute하는 \(F^{n}\) 의 basis \(\mathfrak{B}\) 와\( F^{m}\) 의 basis \(\mathfrak{C}\) 가 존재한다. (따라서, 이 경우에 \(L_{A}\) 와 \(L_{B}\) 는 본질적으로 같은 함수라고 말할 수 있다.)

(관찰 5.5.12)

\(L \in \mathfrak{L}(V,V)\) 이고, \(\mathfrak{B}, \mathfrak{C}\) 가 V의 basis 이면, \(\left[I\right]{\mathfrak{C}}^{\mathfrak{B}} \cdot \left[L\right]{\mathfrak{B}}^{\mathfrak{B}} \cdot \left[I\right]{\mathfrak{B}}^{\mathfrak{C}} = \left[L\right]{\mathfrak{C}}^{\mathfrak{C}}\)

\(A \in \mathfrak{M}{n,n}(F)\) 이고, \(\mathfrak{B}\) 가\( F^{n}\) 의 basis 이면, \(\left[I\right]{\mathfrak{B}}^{\mathcal{E}} \cdot A \cdot \left[I\right]{\mathcal{E}}^{\mathfrak{B}} = \left[L{A}\right]_{\mathfrak{B}}^{\mathfrak{B}}\)

(정의 5.5.13) \(A, B \in \mathfrak{M}{n,n} (F)\) 일 때, 만약 \(U^{-1} A U = B\) 인 invertible matrix \(U \in \mathfrak{M}{n,n} (F)\)가 존재하면, 우리는 \(A ~ B\) 로 표기하고, A similar to B 라고 읽는다.

(관찰 5.5.14) 위 정의의 similarity relation 은 \(\mathfrak{M}_{n,n} (F)\) 의 equivalence relation 이다.

(명제 5.5.15) \(A, B \in \mathfrak{M}_{n,n}(F)\) 일 때, 다음은 동치이다.

\(A ~ B\)

\(\left[L_{A}\right]_{\mathfrak{B}}^{\mathfrak{B}} = B\) 인\( F^{n}\) 의 basis \(\mathfrak{B}\) 가 존재한다.

다음 diagram

%\(\begin{tikzpicture} \matrix (m) \left[matrix of math nodes,row sep=3em,column sep=4em,minimum width=2em\right] { F^{n} & F^{n}

F^{n} & F^{n}

}; \path\left[-stealth\right] (m-1-1) edge node \left[left\right] {\)\alpha_{\mathcal{E}}^{\mathfrak{B}} \(} node \left[right\right] {\)\approx\(} (m-2-1) edge node \left[above\right] {\)L_{A}\(} (m-1-2) (m-1-2) edge node \left[right\right] {\)\alpha_{\mathcal{F}}^{\mathfrak{C}} \(} node \left[left\right] {\)\approx\(} (m-2-2) (m-2-1) edge node \left[below\right] {\)L_{B}\(} (m-2-2);\end{tikzpicture}\)

이 commute 하는 \(F^{n}\) 의 basis \(\mathfrak{B}\) 가 존재한다. (따라서, 이 경우에 \(L_{A}\) 와 \(L_{B}\) 가 본질적으로 같은 함수라고 말할 수 있다.)

(관찰 5.5.16) \(A~B \in \mathfrak{M}_{n,n} (F)\) 이면, 다음이 성립한다.

\(dim ker L_{A} = dim ker L_{B}\).

\(dim im L_{A} = dim im L_{B}\).

\()rk(A) = rk(B)\).

(subsection 5.6) Row-reduced echelon form

(정리 1.2.3 의 재해석)(정리 5.6.2) \(A \in \mathfrak{M}{m,n}(F)\) 이면, \(\left[L{A}\right]_{\mathfrak{C}}^{\mathcal{E}}\) 가 row-reduced echelon form 인 \(F^{m}\) 의 기저 \(\mathfrak{C}\) 가 존재한다. 이 때, A의 row-reduced echelon form 은 유일하게 결정된다.

Determinant 행렬식

(Section 6) 행렬식

(subsection 6.1) Alternating Multilinear form

(정의 6.1.1) \(V_{1}, …, V_{k}\) 가 벡터공간일 때, 함수 \(\mu : V_{1} \times \cdots \times V_{k} \to F\) 가 모든 k-개의 좌표에 대하여 linear 일 때, 즉 임의의\( i = 1,2, …, k\) 에 대하여 \(\mu(…, au_{i} + bw_{i}, …) = a\mu(…, u_{i},…) + b\mu(…,w_{i},….) , (u_{i}, w_{i} \in V_{i}, a,b \in F)\) 일 때, 우리는 \(\mu\) 를 k-linear form 이라고 부른다. 특히, \(k = 2\) 일 때는 \(\mu\) 를 bilinear form 이라고 부른다.

\(V_{1} = \cdots = V_{k} = V\) 인 경우에는 \(\mu\) 를 k-linear form on \(V\) 라고 부른다.

(정의 6.1.6) V 가 벡터공간이고 \(\mu : V \times \cdots \times V \to F\) 가 k-linear form 이라고 하자. 만약 모든 \(v \in V\) 에 대하여 \(\mu (…, v, …, v , …) = 0\) 이면 – 즉 두 좌표가 같을 때 \(\mu\) 의 함수값이 0 이면 – 우리는 \(\mu\) 를 alternating k-linear form on V 라고 부른다.

(관찰 6.1.7) \(\mu : V \times \cdots \times V \to F\) 를 alternating k-linear form 이라고 하면, 모든 \(v, w \in V\) 에 대해 \(\mu(…, v, …, w, …) = - \mu(…, w, …, v, …)\) 이다. 즉, 두 좌표의 위치를 교환하면 부호가 바뀐다.

(표기법 6.1.8) 이 장에서는, \(\mathfrak{M}{n,n} (F) = F^{n} \times \cdots \times F^{n}\) (n 번 곱) 의 표기법을 사용한다. 즉, \(A \in \mathfrak{M}{n,n} (F)\) 일 때,\( A = (\left[A\right]^{1}, …, \left[A\right]^{n})\) (단, \(\left[A\right]^{j}\) 는 A의 j-th column) 으로 표기한다.

(정리 6.1.9) 조건 [\(det(I_{n}) = det (e_{1}, …, e_{n}) = 1\)] 을 만족하는 alternating n-linear form det : \(F^{n} \times \cdots \times F^{n} \to F\) 는 존재하고, 단 하나뿐이다. \(det(A) = |A|\) 의 표기법도 사용한다.

(subsection 6.2) Symmetric Group

(정의 6.2.1) 함수 \(\sigma : \lbrace1, …, n\rbrace \to \lbrace1, …, n\rbrace\) 전체의 집합을 \(T_{n}\) 으로 표기하고, \(S_{n} = \lbrace\sigma \in T_{n} | \sigma is_bijection \rbrace\) 으로 정의한다. \(S_{n}\) 은 symmetric group 이라고 부른다. Symmetric group 의 원소는 permutation 이라고 부른다. 우리는 \(|T_{n}| = n^{n} , |S_{n}| = n!\) 인 것을 잘 알고 있다.

(표기법 6.2.3) \(\sigma \in S_{n}\) 일 때, 우리는 다음의 표기법을 사용한다.

\(\sigma = \begin{pmatrix} 1 & 2 & \cdots & n

\sigma(1) & \sigma(2) & \cdots & \sigma(n) \end{pmatrix}\)

(Cycle notation)\(k \le n\) 이고, \(i_{1}, i_{2}, …, i_{k} (\le n)\) 가 서로 다른 k 개의 자연수일 때, \(\sigma = (i_{1}, i_{2}, …, i_{k}) \in S_{n}\) 은 다음과 같이 정의된 permutation 을 의미한다.

\(\begin{cases} \sigma(i_{h}) = i_{h+1} & (h = 1,2, …, k-1)

\sigma(i_{k}) = i_{1},&

\sigma(j) = j & (j \notin {i_{1}, i_{2}, …, i_{k}}) \end{cases}\)

\((i_{1}, i_{2}, …, i_{k})\) 는 k-cycle 이라고 부른다. 특별히 k = 2 인 경우 \((i_{1}, i_{2})\) 를 transposition 이라고 부른다. (1-cycle 은 항등사상)

(관찰 6.2.8)

\(\sigma\) 와 \(\tau\) 가 disjoint cycle 이면, \(\sigma \bullet \tau = \tau \bullet \sigma\) 이다.

모든 permutation 은 disjoint cycle 들의 합성으로 나타낼 수 있고, 그 방법은 (합성의 순서를 제외하고) 유일하다.

모든 permutation 은 transposition 들의 합성으로 나타낼 수 있다.

(정의 6.2.9) 함수 \(sgn : S_{n} \to {\mp 1}\) 을 \(sgn(\sigma) = (-1)^{r}\) (단, \(\sigma\) 는 r-개 transposition 의 합성) 이라고 정의하자. 우리는 sgn 을 signum 혹은 sign 이라고 읽는다.

(명제 6.2.10) 위 (정의 6.2.9) 의 \(sgn : S_{n} \to {\mp 1}\) 은 well-defined 되어있다.

(정의 6.2.11)\(\sigma \in S_{n}\) 일 때, \(sgn(\sigma) = 1\) 이면 \(\sigma\) 를 even permutation 이라고 부르고, \(sgn(\sigma) = -1\) 이면 \(\sigma\) 를 odd permutation 이라고 부른다. 또, \(A_{n} = {\sigma \in S_{n} | sgn(\sigma) = 1}\) 이라고 표시하고, \(A_{n}\) 을 alternating group 이라고 부른다.

(정의 6.2.15) \(\mathfrak{B} = {v_{1}, …. , v_{n}}\) 이 V의 ordered basis 이고 \(\sigma \in S_{n}\) 일 때, V의 새로운 ordered basis \(\mathfrak{B}{\sigma} = {v{\sigma(1)} , v_{\sigma(2)}, …, v_{\sigma(n)}}\) 을 생각하자. 이때, (\(n \times n\))-transition matrix \(\left[I\right]{\mathfrak{B}}^{\mathfrak{B}{\sigma}}\) 를 \(\sigma\) 에 대응하는 permutation matrix 라고 부르고, \(I_{\sigma} = \left[I\right]{\mathfrak{B}}^{\mathfrak{B}{\sigma}}\) 로 표기한다.

(관찰) \(\sigma, \tau \in S_{n}\) 일 때, 다음이 성립한다.

\(I_{\sigma}\) 의 j-th column 은 \(e_{\sigma(j)}\) . 즉 \(I_{\sigma} = (e_{\sigma(1)}, …, e_{\sigma(n)}) \)

\(A \in \mathfrak{M}{n,n}(F)\) 이면, \(AI{\sigma} = (\left[A\right]^{\sigma(1)} , …, \left[A\right]^{\sigma(n)})\) .

\(I_{\sigma} \cdot I_{\tau} = I_{\sigma \bullet \tau}\).

\((I_{\sigma})^{-1} = I_{(\sigma^{-1})} = ^{t}(I_{\sigma}) \)

\(I_{\sigma} = I_{\tau}\) 이면, \(\sigma = \tau\)

(정의 6.2.19) \(\mathfrak{B} = {v_{1}, …, v_{n}}\) 이 V의 고정된 ordered basis 이고 \(\sigma \in S_{n}\) 일 때, Linear extension theorem 을 이용하여 선형사상 \(P_{\sigma} : V \to V\) 를 \(P_{\sigma} (v_{i}) = v_{\sigma(i)}\) , (i = 1, …, n) 으로 정의하자.

(표기법 6.2.22) \(\lambda_{i} \in F\) 일 때, (i,i)-성분이 \(\lambda_{i}\) 인 (\(n \times n\))-대각행렬 (diagonal matrix) 를 \(diag(\lambda_{1}, …, \lambda_{n})\) 으로 표기한다.

(subsection 6.3) Determinant 의 정의 1

(관찰 6.3.1) 만약 \(D : \mathfrak{M}{n,n} (F) \to F\) 가 (정리 6.1.9) 의 조건을 만족한다면, \(A \in \mathfrak{M}{n,n}(F)\)이고 \(\sigma \in S_{n}\) 일 때, 다음이 성립한다.

\(D(\left[A\right]^{\sigma(1)} , …, \left[A\right]^{\sigma(n)} ) = sgn(\sigma) \cdot D(\left[A\right]^{1}, …, \left[A\right]^{n}) = sgn(\sigma) \cdot D(A)\),

\(D(e_{\sigma(1)} , … e_{\sigma(n)}) = sgn(\sigma)\)

(정리 6.3.2. 정리 6.1.9 의 \(det : \mathfrak{M}{n,n} (F) \to F\) 는 \(det(A) = |A| = det(a{ij}) = \sum_{\sigma \in S_{n}} sgn(\sigma) \cdot a_{\sigma(1), 1} a_{\sigma(2), 2} \cdots a_{\sigma(n), n}\) 으로 주어진다.

(따름정리 6.3.7) \(A \in \mathfrak{M}_{n,n} (F)\) 이면 \(det(^{t}A) = det(A)\) 이다.

(따름정리 6.3.8) \(det : \mathfrak{M}_{n,n} (F) \to F\) 를 n-개의 row들에 관한 함수로 보더라도, det 는 \(det(I) = 1\)인 유일한 alternating n-linear form 이다.

(subsection 6.4) Determinant 의 성질

(명제 6.4.1) \(A, B \in \mathfrak{M}_{n,n} (F)\) 이면, \(det(AB) = det(A)det(B)\) 가 성립한다.

(따름정리 6.4.2) A가 invertible이면, 물론 \(det(A) \neq 0\) 이고, \(det(A^{-1}) = \frac{1}{det(A)} = det(A)^{-1}\) 이다.

(관찰 6.4.5) \(A \in \mathfrak{M}_{n,n} (F)\)일 때, 만약 \({\left[A\right]^{1}, …, \left[A\right]^{n}}\) 이 linearly dependent 이면, \(det(A) = 0\) 이다.

(따름정리 6.4.6) 다음은 동치이다.

A는 invertible.

\(det(A) \neq 0\).

(Gaussian Elimination 과 Determinant) (명제 6.4.11) 행렬식은 다음 성질을 갖는다.

어떤 한 column 에 다른 column 의 상수배를 더해주어도, 행렬식은 변하지 않는다. 즉, \(i \neq j\) 이고 \(c \in F\) 일 때, i-th column 에 [j-th column 의 c-배] 를 더해주면, \(det(…, \left[A\right]^{i} + c\left[A\right]^{j}, …, \left[A\right]^{j}, …) = det(A)\) 이다.

위 성질에서 ‘column’ 을 ‘row’로 바꾸어도 성질이 성립한다.

(subsection 6.5) Determinant 의 정의 2

(표기법 6.5.1) \(A = (a_{ij}) \in \mathfrak{M}{n,n}(F)\) 일 때, \(M{ij}\) 를 [A에서 i-th row 와 j-th column 을 제거한] (\((n-1) \times (n-1)\))-행렬이라 하고, \(\hat{A}{ij} = det(M{ij})\) 로 표기한다. 우리는 \(\hat{A}_{ij}\) 를 A 의 (i,j)-minor 라 부른다.

(정의 6.5.2) j-th column 에 관한 전개(expansion) \(D^{j} : \mathfrak{M}{n,n} (F) \to F\) 를 \(D^{j}(A) = \sum{i =1}^{n} (-1)^{i+j} a_{ij}\hat{A}{ij} , (j = 1,…,n)\) 로 정의하고, 마찬가지로 i-th row 에 관한 expansion \(D{i} : \mathfrak{M}{n,n} (F) \to F\) 를 \(D{i}(A) = \sum_{j =1}^{n} (-1)^{i+j} a_{ij}\hat{A}_{ij} , (j = 1,…,n)\) 로 정의한다.

(정리 6.5.6) (정의 6.5.2) 에서 정의된 2n-개의 함수 \(D_{i}, D^{j} : \mathfrak{M}{n,n}(F) \to F\) 는 모두 alternating n-linear form 이고, \(I{n}\) 에서의 값이 1이다. 따라서, 이 2n-개의 함수가 모두 \(det : \mathfrak{M}_{n,n}(F) \to F\) 와 같은 함수이다.

(subsection 6.6) Cramer’s Rule

(Cramer’s Rule) (정리 6.6.1) \(A \in \mathfrak{M}{n,n}(F)\) 일 때, \(X = ^{t}(x{1}, …, x_{n}) \in F^{n}\) 이 연립방정식 \(AX = B\) 의 solution이면, \(det(A)x_{i} = det(\left[A\right]^{1}, …, B, …, \left[A\right]^{n})\) (i = 1, …, n) 이어야 한다. 이 식에서 \((\left[A\right]^{1}, …, B, …, \left[A\right]^{n}\)) 은 A의 i-tho column 을 B로 바꿔치기한 행렬을 뜻한다.

(명제 6.6.4) \(A = (a_{ij}) \in \mathfrak{M}{n,n}(F)\) 가 invertible matrix 이면,\( A^{-1} = ^{t} (\frac{(-1)^{i+j} \hat{A}{ij}}{det(A)} ) \)이다. 즉, \(A^{-1}\) 의 (i,j)-좌표는 \(\frac{(-1)^{i+j} \hat{A}_{ji}}{det(A)}\) 이다.

(subsection 6.7) Adjoint Matrix

(정의 6.7.1) \(A \in \mathfrak{M}{n,n}(F)\) 일 때, \(adj(A) = ^{t}((-1)^{i+j} \hat{A}{ij})\) (즉 , \(adj(A)\) 의 (i,j)-좌표는 \((-1)^{i+j} \hat{A}_{ji})\) 라고 표기하고, \(adj(A)\) 를 A의 adjoint matrix 라 부른다.

(정리 6.7.2) \(A \in \mathfrak{M}{n,n} (F)\) 이면, \(A \cdot adj(A) = det(A) \cdot I{n} = adj(A) \cdot A\) 이다. (A 가 가역일 필요는 없다.)

(따름정리 6.7.3) \(A = (a_{ij}) \in \mathfrak{M}{n,n}(F)\) 이면, \(\sum{k = 1}^{n} (-1)^{k+j} a_{ik} \hat{A}{jk} = \delta{ij} \cdot det(A) = \sum_{k=1}^{n} (-1)^{k+j} a_{ki} \hat{A}_{kj}\)

Characteristic polynomial and Diagonalization 특성다항식과 대각화

(section 7) 특성다항식과 대각화

(subsection 7.1) Eigen-vector 와 Eigen-value

(정의 7.1.1)\(A \in \mathfrak{M}_{n,n}(F)\) 일 때, \(AX = \lambda X\) 인 \(\lambda \in F\) 와 \(0 \neq X \in F^{n}\) 이 존재하면, 우리는 X를 eigen-value \(\lambda\) 를 갖는 (\(\lambda\) 에 대응하는) A의 eigenvector라 부른다. 마찬가지로, \(L \in \mathfrak{L}(V,V)\) 이고, \(Lv = \lambda v\) 인 \(\lambda \in F\) 와 \(0 \neq v \in V\) 가 존재할 때, 우리는 v 를 eigen-value \(\lambda\) 를 갖는 L의 eigen-vector 라 부른다.

(표기법 7.1.2) 앞으로 [\(T \in \mathfrak{L}\mathfrak{M}\)] 이라고 하면, 항상 [\(T \in \mathfrak{L}(V,V)\) 또는 \(T \in \mathfrak{M}{n,n}(F)\)] 를 뜻하기로 약속한다. 이 때, \(dim V = n\) 이고, vector들은 u,v,w 등으로 표기하며, \(I{V} = I_{n} = I\) 는 혼동하기로 한다. 물론 T가 행렬이면 \(V = F^{n}\) 으로 이해하고, \(T = L_{T} \)로 혼동한다. (따라서, \(A \in \mathfrak{M}{n,n}(F)\)) 일 때, \(ker A = ker L{A}\) 의 표기법도 가능.) 또, 혹시 모르니, \(V \neq 0\) 이라고 가정.

(재정의 7.1.3) \(T \in \mathfrak{L} \mathfrak{M}\) 일 때, \(Tv = \lambda v\) 인 \(\lambda \in F\) 와 \(0 \neq v \in V \)가 존재하면, 우리는 v를 eigen-value \(\lambda\) 를 갖는 T의 eigen-vector 라고 부른다.

(정의 7.1.7) \(L \in \mathfrak{L} (V,V)\) 일 때, V의 임의의 기저 \(\mathfrak{B}\) 를 골라 \(det(L) = det(\left[L\right]_{\mathfrak{B}}^{\mathfrak{B}})\) 로 정의하자.

(관찰 7.1.8) (정의 7.1.7) 의 \(det(L)\) 은 well-defined 되어있다.

(정의 7.1.9) \(T \in \mathfrak{L} \mathfrak{M}\) 이고, [T 혹은 \(\left[T\right]{\mathfrak{B}}^{\mathfrak{B}} \)의 좌표] 를 \(a{ij}\) 로 표기할 때, T 의 characteristic polynomial (특성다항식) \(\phi_{T}(t) \in F\left[t\right]\) 를 \(\phi_{T}(t) = det(tI – T) = \begin{vmatrix} t-a_{11} & -a_{12} & -a_{13} & \cdots & -a_{1n}

-a_{21} & t-a_{22} & -a_{23} & \cdots & -a_{2n}

\vdots & \vdots & \vdots & \ddots & \vdots

-a_{n1} & -a_{n2} & -a_{n3} & \cdots & t-a_{nn}

\end{vmatrix}\)

으로 정의한다. (이 때, \(tI – T\) 의 t는 마치 scalar 인 것처럼 생각한다.)

(관찰 7.1.10) Characteristic polynomial 은 similar matrix 의 invariant 이다. 따라서, \(T \in \mathfrak{L} \mathfrak{M}\)일 때, \(\phi_{T}(t)\) 는 well-defined 된다.

(명제 7.1.11) \(T \in \mathfrak{L} \mathfrak{M}\) 일 때, \(\lambda \in F \)가 T의 eigen-value 이기 위한 필요충분조건은 \(\phi_{T}(\lambda) = 0\) 인 것이다.

(관찰 7.1.12) Trace, rank, determinant, characteristic 및 eigen-value 는 similar matrix 의 invariant 이다.

(정리 7.1.18) 모든 다항식 \(f(t) \in \mathbb{C}\left[t\right]\) 는 (\(\mathbb{C}\) 에서) 근 (root) 를 갖는다. (즉, 인수정리에 의해, \(f(t) \in \mathbb{C}\left[t\right]\) 는 \(\mathbb{C}\) 위의 일차식들의 곱으로 인수분해된다.)

(subsection 7.2.) Diagonalization

(정의 7.2.1) \(A \in \mathfrak{M}{n,n} (F)\) 일 때, \(A ~ D\) 인 diagonal matrix \(D \in \mathfrak{M}{n,n} (F)\) 가 존재하면, A를 diagonalizable matrix 라 부른다.

(관찰 7.2.2) \(A \in \mathfrak{M}_{n,n}(F)\) 일 때, 다음 조건들은 동치이다.

A는 diagonalizable.

A 의 eigen-vector 들로 이루어진 \(F^{n}\) 의 basis 가 존재.

(정의 7.2.3) Linear operator \(L \in \mathfrak{L}(V,V)\) 의 (V의 임의의 기저 \(\mathfrak{B}\) 에 대한) 행렬 \(\left[L\right]_{\mathfrak{B}}^{\mathfrak{B}}\) 가 diagonalizable 이면, L 도 diagonalizable 이라고 말한다.

(관찰 7.2.4) Linear operator \(L \in \mathfrak{L} (V,V)\) 가 diagonalizable 이기 위한 필요충분조건은\( \left[L\right]_{\mathfrak{B}}^{\mathfrak{B}}\) 가 diagonalizable matrix 인 V의 기저 \(\mathfrak{B}\) 가 존재하는 것이다.

(재정의 7.2.5) \(T \in \mathfrak{L} \mathfrak{M}\) 일 때, \(\left[T\right]_{\mathfrak{B}}^{\mathfrak{B}}\) 가 diagonal matrix 인 V 의 기저 \(\mathfrak{B}\) 가 존재하면 – 즉, T의 eigen-vector 들로 이루어진 V의 기저 \(\mathfrak{B}\) 가 존재한다면 – T는 diagonalizable 이라 말한다.

(명제 7.2.6) \(T \in \mathfrak{L} \mathfrak{M}\) 일 때, \(\phi_{T}(t)\) 가 F에서 서로 다른 n-개의 root 를 가지면, T 는 diagonalizable이다. (물론, \(dim V = n\))

(보조정리 7.2.7) \(T \in \mathfrak{L} \mathfrak{M}\) 일 때, \(v_{1}, …, v_{k}\) 가 T의 eigen-vector 라고 하자. 만약, \(v_{i} 의 eigen-value \lambda_{i}\) 들이 mutually distinct (즉, \(\lambda_{i} \neq \lambda_{j} if i \neq j\)) 이면, \({v_{1}, …, v_{k}}\) 는 일차독립이다.

(subsection 7.3.) Caley-Hamilton Theorem

(정의 7.3.2) \(T \in \mathfrak{L} \mathfrak{M}\) 일 때, \(\mathcal{I}_{T} = {f(t) \in F\left[t\right] | f(T) = 0}\) 라고 정의하자.

(명제 7.3.4) \(T \in \mathfrak{L} \mathfrak{M}\) 이면, \(\mathcal{I}_{T} \neq 0\) 이다.

(Cayley-Hamilton Theorem) (정리 7.3.5) \(T \in \mathfrak{L} \mathfrak{M}\) 이면, \(\phi_{T}(T) = 0\) 이다. 즉, \(\phi_{T}(t) \in \mathcal{I}_{T}\) 이다.

(subsection 7.4) Minimal Polynomial

(정의 7.4.1) \(T \in \mathfrak{L} \mathfrak{M}\) 일 때, \(\mathcal{I}{T}\) 의 non-zero polynomial 중에서 [최소의 degree 를 갖는 monic polynomia] 을 T의 minimal polynomial 이라고 부르고, \(m{\tau}(t)\) 로 표기한다.

(관찰 7.4.2) \(A, B \in \mathfrak{M}{n,n}(F)\) 이고 \(A ~ B\) 이면, \(m{A}(t) = m_{b}(t)\) 이다. 즉, minimal polynomial 은 similar matrix 의 invariant 이다.

(정리 7.4.3) \(T \in \mathfrak{L}\mathfrak{M}\) 일 때, T의 minimal polynomial 은 존재하고 하나뿐이다.

(따름정리 7.4.4)\(T \in \mathfrak{L}\mathfrak{M}\) 일 때, \(f(t) \in \mathcal{I}{T}\) 이면, \(f(t)\) 는 \(m{T}(t)\) 의 배수이다.

특별히 , \(\phi_{T}(t)\) 도 \(m_{T}(t)\) 의 배수이다.

(subsection 7.5) Direct sum 과 Eigen-space Decomposition

(정의 7.5.1) V가 벡터공간이고 U,W 가 V의 부분공간이라고 하자. 이 때, \(U \cap W\) 이면, \(U + W = U \oplus W \)로 표기하고, “\(U+W\) 는 \(U,W\) 의 direct sum이다” 라고 말한다.

(관찰 7.5.2) V가 vector space 이고, U,W 가 V 의 subspace 라고 할 때, 다음 조건은 동치이다.

\(V = U \oplus W,\)

\(V = U + W\) 이고 \(U \cap W = 0\)

이 떄, U 를 W의 direct complement 라고 부른다.

(관찰 7.5.3) V가 vector space 이고, U,W 가 V의 subspace 라고 할 때, 다음은 동치이다.

\(V = U \oplus W\)

V 의 모든 vector v 는 \(u + w\) (단, \(u \in U\), \(w \in W\)) 의 꼴로 쓸 수 있고, 그 방법은 하나뿐이다.

\(V = U + W\) 이고, V의 zero vector 0 를 \(u + w\) (단, \(u \in U\), \(w \in W\) ) 의 꼴로 쓰는 방법은 하나뿐이다.

(관찰 7.5.4.) V가 f.d.v.s. 이고, \(U, W \le V\) 일 때, 다음 조건은 동치이다.

\(V = U \oplus W,\)

\(dim V = dim U + dim W\) 이고, \(U \cap W = 0.\)

(관찰 7.5.6.) V가 벡터공간이고 \(\mathfrak{B}{1}, \mathfrak{B}{2}\) 가 각각 V의 subspace \(W_{1}, W_{2}\) 의 기저라고 하면, 다음 조건은 동치이다.

\(V = W_{1} \oplus W_{2}\).

\(\mathfrak{B} = \mathfrak{B}{1} \mathfrak{U} \mathfrak{B}{2}\) 는 V의 basis. (단, \(\mathfrak{U}\) 는 disjoint union 을 의미한다.)

(정의 7.5.7) V가 vector space 이고, \(W_{1}, …, W_{k}\) 가 V의 subspace 라고 하자. 이 때, \(W_{1} + \cdots + W_{k}\) 의 vector v를 \(w_{1}, + \cdots + w_{k}\) (단, \(w_{i} \in W_{i}\)) 의 꼴로 표현하는 방법이 하나뿐이면, \(\sum_{i = 1}^{k} W_{i} = W_{1} + \cdots + W_{k} = W_{1} \oplus \cdots \oplus W_{k} = \bigoplus_{i = 1}^{k} W_{i}\) 로 표기하고, 이런 경우에 “\(\sum_{i = 1}^{k} W_{i}\) 는 \(W_{1}, …, W_{k}\) 의 direct sum 이다” 라고 말한다. 또, 각각의 \(W_{1}, …, W_{k}\) 는 \(\oplus_{i=1}^{k}\) 의 direct summand 라고 말한다.

(관찰 7.5.9) V가 vector space 이고 \(W_{1}, …, W_{k}\) 가 V의 subspace 라고 할 때, 다음은 동치이다.

\(V = W_{1} \oplus \cdots \oplus W_{k}\) .

V 의 모든 vector v 는 \(w_{1} + \cdots + w_{k}\) (단, \(w_{i} \in W_{i}\)) 의 꼴로 쓸 수 있고, 그 방법은 하나뿐이다.

\(V = W_{1} + \cdots + W_{k}\) 이고, V의 zero vector 0 을 \(w_{1} + \cdots + w_{k}\) (단, \(w_{i} \in W_{i}\)) 의 꼴로 쓰는 방법은 하나 뿐이다.

(관찰 7.5.11) \(\mathfrak{B}{i}\) 가 각각 V의 subspace \(W{i}\) (단, \(i = 1,…,k\)) 의 기저라고 할 때, 다음 조건은 동치이다.

\( V = W_{1} \oplus \cdots \oplus W_{k}\)

\(\mathfrak{B} = \mathfrak{B}{1} \mathfrak{U} \cdots \mathfrak{U} \mathfrak{B}{k}\) 는 V의 basis. (이 때, \(\mathfrak{U}\) 는 mutually disjoint union 을 의미한다. 즉\( \mathfrak{B}{i} \cap \mathfrak{B}{j} = \empty if i \neq j\) 인 union 이라는 뜻이다.)

(정의 7.5.15.) \(T \in \mathfrak{L}\mathfrak{M}\)이고, \(\lambda \in F\) 일 때, \(V_{\lambda} = V_{T,\lambda} = {v \in V | Tv = \lambda v}\) 로 표기하고, \(V_{\lambda} = V_{T,\lambda}\) 를 \(\lambda\) 에 대응하는 T의 eigen-space 라고 부른다.

(관찰 7.5.17) \(T \in \mathfrak{L}\mathfrak{M}\) 일 때, 다음은 동치이다.

T는 diagonalizable.

\(V = V_{\lambda_{1}} \oplus \cdots \oplus V_{\lambda_{k}}\) 인, T의 (서로 다른) eigen-value \(\lambda_{1}, …, \lambda_{k}\) 존재.

(정의 7.5.18) 위 (관찰 7.5.17) 의 \(V = V_{\lambda_{1}} \oplus \cdots \oplus V_{\lambda_{k}}\) 를 diagonalizable T 에 관한 (V의) eigen-space decomposition 이라고 부른다.

Decomposition theorem 분해정리

(Section 8) 분해정리

(subsection 8.1) polynomial

(명제 8.1.4) \(F\left[t\right]\) 의 polynomial \(f_{1}(t), f_{2}(t), … , f_{k}(t)\) 의 최대공약수를 \(d(t)\) 라고 하면, \(d(t) = g_{1}(t) f_{1}(t) g_{2}(t) f_{2}(t) + \cdots + g_{k}(t) f_{k}(t)\) 가 성립하는 \(F\left[t\right]\) 의 polynomial \(g_{1}(t), g_{2}(t), … , g_{k}(t)\) 가 존재한다.

(명제 8.1.5) \(T \in \mathfrak{L}\mathfrak{M}\) 의 characteristic polynomial \(\phi_{T}(t)\) 와 minimal polynomial \(m_{T}(t)\) 의 [F-위의 monic irreducible divisor 의 집합] 은 같다.

(표기법 8.1.10)\(f(t) \in \mathbb{C} \left[t\right]\) 가\( f(t) = \alpha_{n}t^{n} + \alpha_{n-1}t^{n-1} + \cdots + \alpha_{1}t + \alpha_{0}\) (단, \(\alpha_{0} , …, \alpha_{n} \in \mathbb{C}\)) 로 주어졌을 때, \(\hat{F}(t) \in \mathbb{C} \left[t\right] 를 \hat{f}(t) = \hat{\alpha_{n}} t^{n} + \hat{\alpha_{n-1}} t^{n-1} + \cdots + \hat{\alpha_{1}} t + \hat{\alpha_{0}}\) 으로 정의한다. 또, \(A = (\alpha_{ij}) \in \mathfrak{M}{m,n}(\mathbb{C})\) 일 때, \(\hat{A} = (\hat{\alpha{ij}}) \in \mathfrak{M}_{m,n} (\mathbb{C})\) 로 정의한다.

(관찰 8.1.14.) \(A \in \mathfrak{M}_{n,n}(\mathbb{R})\) 을 real matrix 로 생각하나 complex matrix 로 생각하나 characteristic polynomial 과 minimal polynomial 은 변함이 없다.

(subsection 8.2) T -invariant Subspace

(정의 8.2.1) \(T \in \mathfrak{L}\mathfrak{M}\) 이고 \(W \le V\) 일 때, \(T(W) \le W\) 이면 – 즉 \(T |{W} : W \to W\) 가 의미가 있으면 – 우리는 W를 V의 T-invatiant subspace 라고 부른다. 혹은, W 는 “T-stable” 이다 라고 말한다. (만약, \(T \in \mathfrak{M}{n,n}(F)\) 이면, \(T = L_{T}, V = F^{n}\)으로 이해한다.)

(관찰 8.2.9) \(T \in \mathfrak{L}\mathfrak{M}\) 이고 \(f(t) \in F\left[t\right]\) 라고 하면 다음이 성립한다.

ker T 와 im T 는 T-invariant.

ker f(T) 와 im f(T) 는 T-invariant.

(subsection 8.3) Primary Decomposition Theorem

(표기법 8.3.3) \(T \in \mathfrak{L}\mathfrak{M}\) 일 때, \(\phi_{T}(t) = p_{1}(t)^{e_{1}} p_{2}(t)^{e_{2}} \cdots p_{k}(t)^{e_{k}}, m_{T}(t) = p_{1}(t)^{f_{1}} p_{2}(t)^{f_{2}} \cdots p_{k}(t)^{f_{k}}\) 로 (F-위에서) 인수분해된다고 하자. 이 때 \(p_{i}(t)\) 들은 F[t] 의 relatively prime monic irreducible polynomial 이고 \(1 \le f_{i} \le e_{i}\) 이다. 또, \(W_{i} = ker p_{i}(T)^{e_{i}} , T_{i} = T|{W{i}} , (i = 1,…,k)\) 로 간단히 표기하기로 한다. \(W_{i}\) 가 T-invariant subspace 이다.

(Primary Decomposition Theorem) (정리 8.3.4) \(T \in \mathfrak{L}\mathfrak{M}\) 이면, \(V = ker p_{1}(T)^{e_{1}} \oplus ker p_{2} (T)^{e_{2}} \oplus \cdots \oplus ker p_{k}(T)^{e_{k}} = ker p_{1}(T)^{f_{1}} \oplus ker p_{2}(T)^{f_{2}} \oplus \cdots \oplus ker p_{k}(T)^{f_{k}}\) 로 분해할 수 있다. 그리고 모든 \(i = 1, …, k\) 에 대해, 다음이 성립한다.

\(W_{i} = ker p_{i}(T)^{e_{i}} = ker p_{i}(T)^{f_{i}}\).

\(m_{T_{i}}(t) = p_{i}(t)^{f_{i}}\) .

\(\phi_{T_{i}}(t) = p_{i}(t)^{e_{i}}\) . (따라서, \(dim W_{i} = e_{i} \cdot deg(p_{i})\).)

(보조정리 8.3.5.) \(T \in \mathfrak{L}\mathfrak{M}\) 일 때,\( f(t), g(t) \in F\left[t\right]\) 는 monic 이고 relatively prime 이라고 하자. 만약 \(\xi(t) = f(t) g(t) \in \mathcal{I}_{T}\) 이면, \(V = ker f(T) \oplus ker g(T)\) 로 쓸 수 있다. 이때, \(U = ker f(T), W = ker g(T) \)로 간단히 표기하면,

\(\xi(t) = m_{T} (t)\) 일 때에는, \(m_{T|v}(t) = f(t)\) 이고 \(m_{T|w} (t) = g(t)\)

\(\xi(t) = \phi_{T}(t)\) 일 때에는, \(\phi_{T|v} (t) = f(t)\) 이고 \(\phi_{T|w} (t) = g(t)\) .

(subsection 8.4.) Diagonalizability

(따름정리 8.4.1.) \(T \in \mathfrak{L}\mathfrak{M}\) 이 diagonalizable일 필요충분조건은 T의 minimal polynomial \(m_{T}(t)\) 가 (F-위에서) 일차식들로 인수분해되고 multiple root 을 갖지 않는 것이다. (즉, 모든\( i = 1,…,k\) 에 대하여, \(deg(p_{i}) = 1\) 이고, \(f_{i} = 1\) 인 것이다.)

(따름정리 8.4.2.) \(T \in \mathfrak{L}\mathfrak{M}\) 이 diagonalizable 이고 W가 V의 T-invariant sub-space 이면, \(T|_{W}\) 도 diagonalizable 이다.

(정의 8.4.3.) \(T , S \in \mathfrak{L}\mathfrak{M}\) 일 때, V의 하나의 기저 \(\mathfrak{B}\) 에 관해 \(\left[T\right]{\mathfrak{B}}^{\mathfrak{B}}\) 와 \(\left[S\right]{\mathfrak{B}}^{\mathfrak{B}}\) 가 모두 대각행렬이면, 우리는 T와 S가 simultaneously diagonalizable 하다고 말한다. 여러 개의 linear operator 의 경우에도 마찬가지로 정의한다.

(명제 8.4.4.) \(T , S \in \mathfrak{L}\mathfrak{M}\)이고, 조건 \(TS = ST\) 를 만족한다고 하자. 이 때, T와 S가 각각 diagonalizable 이면 T,S 는 simultaneously diagonalizable 이다.

(subsection 8.5) T-Cyclic Subspace

(정의 8.5.1.) \(T \in \mathfrak{L}\mathfrak{M}\) 일 때, \(V = \lbracef(T) v | f(t) \in F\left[t\right]\rbrace\) 인 \(v \in V\) 가 존재하면, V를 T-cyclic space 라 부르고, \(V = F\left[t\right] v = \lbracef(T) v | f(t) \in F\left[t\right]\rbrace\) 로 표기한다.

(명제 8.5.4.) \(T \in \mathfrak{L}\mathfrak{M} V = F\left[t\right] v\) 가 T-cyclic 이면, 다음이 성립한다.

\(\phi_{T}(t) = m_{T}(t).\)

\(\mathfrak{B} = {v, Tv, T^{2}v, …, T^{n-1} v}\) 는 V의 기저. 단,\( n = dim V = deg (m_{T})\) .

\(\left[T\right]{\mathfrak{B}}^{\mathfrak{B}}\) 는 \(m{T}(t)\) 에 대응하는 companion matrix.

(정의 8.5.7.) \(T \in \mathfrak{L}\mathfrak{M}\) 이고, W는 V의 T-invariant subspace 라고 하자. 이 때 W가 \((T|_{W})\)-cyclic 이면 , W 를 V의 T-cyclic subspace 라고 부른다.

(정의 8.5.9.) \(T \in \mathfrak{L}\mathfrak{M}\) 이고, \(0 \neq w \in V\) 일 때, \(F\left[t\right] w = {f(T) w | f(t) \in F\left[t\right]} = \langle w , Tw , T^{2}w, … \rangle\) 로 표기하고, \(F\left[t\right] w\) 를 =[T-cyclic subspace of V generaged by w] 라고 부른다.

\(W = F\left[t\right] w\) 로 놓을 때, \(m_{w}(t) = m_{(T|{W})} (t)\) 로 간략히 표기하고, \(m{w}(t)\) 를 [minimal polynomial of w in V] 라고 부른다.

(관찰 8.5.11.) \(T \in \mathfrak{L}\mathfrak{M}\) 이고, \(0 \neq w \in V\) 일 때, \(W = F\left[t\right]w\) 라고 놓으면, 다음이 성립한다.

\( m_{w}(T) = \phi_{T|w} (t)\).

\(\mathfrak{C} = {w, Tw, …, T^{m-1} w}\) 는 W의 기저. 단,\( m = dim W = deg(m_{w})\) .

\(\left[T|{W}\right]{\mathfrak{C}}^{\mathfrak{C}}\) 는 \(m_{w}(t)\) 에 대응하는 companion matrix.

\(m_{T} (t)\) 는 \(m_{w} (t)\) 의 배수.

(subsection 8.6) Cyclic Decomposition Theorem

(Cyclic Decomposition Theorem) (정리 8.6.1.) \(T \in \mathfrak{L}\mathfrak{M}\) 일 때, \(m_{T} (t) = p(t)^{f}\) 라고 가정하자. 이 때 \(p(t)\) 는 \(F\left[t\right]\) 의 monic irreducible polynomial 이다. 그러면, \(V = U_{1} \oplus U_{2} \oplus \cdots \oplus U_{h}\) 인 V의 (T-invariant) T-cyclic subspace \(U_{1}, …, U_{h}\) 가 존재한다. 그리고, \(\phi_{T|{U{j}}}(t) = m_{T|{U{j}}}(t) = p(t)^{r_{j}} (j = 1,…,h)\) 로 표기할 떄, 다음 조건 \(f = r_{1} \ge r_{2} \ge \cdots \ge r_{h} \ge 1\) 을 만족하는 자연수 h 와 \(r_{1} , … , r_{h}\) 는 유일하게 결정된다.

(따름정리 8.6.2) 두 square matrix \(A, B \in \mathfrak{M}{n,n}(F)\) 에 대해, \(A ~ B\) 인지 여부를 결정해주는 invariant 들의 집합은 \({p{i}(t), h_{i}, r_{ij}}\( 이다.

Rigid Motion

(Section 9) \mathbb{R}^{n} 의 Rigid Motion

(subsection 9.1) \mathbb{R}^{n} -공간의 Dot product 와 Euclidean norm

(정의 9.1.1) X = ^{t}(a_{1} , …, a_{n}) , Y = ^{t}(b_{1}, …, b_{n}) \in \mathbb{R}^{n} 일 때, \langle X , Y \rangel = ^{t} X \cdot Y = \sum_{i = 1}^{n} a_{i}b_{i} 로 정의하고 이를 \mathbb{R}^{n} 의 dot product 라고 부른다. 그리고 dot product 가 주어진 \mathbb{R}^{n}-공간을 Euclidean space 라고 부른다.

(정의 9.1.4) X \in \mathbb{R}^{n} 일 때, \Vert X \Vert = \sqrt{\langle X , X \rangle} 로 표기하고, \Vert X \Vert 를 X의 (Euclidean) norm 이라고 부른다. Norm 은 vector 의 길이(크기)로 해석한다.

(정의 9.1.9.) X,Y \in \mathbb{R}^{n} 일 때, \langle X, Y \rangle = 0 이면, X \perp Y 로 표기하고 X 와 Y는 서로 수직 (perpendicular, 또는 orthogonal) 이라고 말한다.

S, T \subseteq \mathbb{R}^{n} 일 때, [\langle X , Y \rangle = 0 for all X \in S , Y \in T ] 이면, S \perp T 로 표기하고 S와 T는 서로 수직이라고 말한다.

(정의 9.1.11) \mathbb{R}^{n} 의 non-zero vector 들 X_{1}, …, X_{m} 이 mutually perpendicular 이면 – 즉 , [X_{i} \perp X_{j} for all 1 \le i \neq j \le m] 이면 – {X_{1}, …, X_{m}} 을 \mathbb{R}^{n} 의 orthogonal subset 이라고 부른다. 이 때, [\Vert X_{k} \Vert = 1 for all 1 \le k \le m] 의 조건도 만족하면, {X_{1}, …, X_{m}} 을 \mathbb{R}^{n} 의 orthonormal subset 이라고 부른다.

\mathbb{R}^{n} 의 basis \mathfrak{B} 가 orthogonal subset 이면, \mathfrak{B} 를 \mathbb{R}^{n} 의 orthogonal basis 라고 부르고, orthonormal subset 이면 \mathfrak{B} 를 \mathbb{R}^{n} 의 orthonormal basis 라고 부른다.

(정의 9.1.14) S \subseteq \mathbb{R}^{n} 일 때, S^{\perp} = {X \in \mathbb{R}^{n} | X \perp Y for all Y \in S } 로 표기하고, “S perp” 로 읽는다. 특히 W \le \mathbb{R}^{n} 일 때에는 W^{\perp} 를 W의 orthogonal complement 라고 부른다.

(관찰 9.1.15) S \subseteq \mathbb{R}^{n} , W \le \mathbb{R}^{n} 이고 \mathfrak{C} 가 W의 basis이면, 다음이 성립한다.

S^{\perp} 는 \mathbb{R}^{n} 의 subspace.

S^{\perp} = \langle S \rangle^{\perp}.

W^{\perp} = \mathfrak{C}^{\perp}.

(subsection 9.2.) \mathbb{R}^{n} -공간의 Rigid Motion

(정의 9.2.1.) 함수 M : \mathbb{R}^{n} \to \mathbb{R}^{n} 이 조건 \Vert M(X) – M(Y) \Vert = \Vert X – Y \Vert (X,Y \in \mathbb{R}^{n}) 을 만족하면 M을 \mathbb{R}^{n} 의 rigid motion 또는 isometry 라고 부른다.

(관찰 9.2.5) 함수 M : \mathbb{R}^{n} \to \mathbb{R}^{n} 이 \mathbb{R}^{n} 의 rigid motion이면,

M은 injective 이다.

M은 연속함수이다.

(관찰 9.2.8.) L(0) = 0 인 \mathbb{R}^{n} 의 rigid motion L 은 다음 성질을 갖는다.

\Vert L(X) \Vert = \Vert X \Vert for all X \in \mathbb{R}^{n}.

\langle L(X), L(X) \rangle = \langle X , X \rangle for all X \in \mathbb{R}^{n}.

\langle L(X), L(Y) \rangle = \langle X, Y \rangle for all X , Y \in \mathbb{R}^{n}.

\mathfrak{B} 가 \mathbb{R}^{n} 의 orthonormal basis 이면, L(\mathfrak{B}) 도 \mathbb{R}^{n} 의 orthonormal basis.

(관찰 9.2.9) A \in \mathfrak{M}{n,n}(\mathbb{R}) 의 column 들 {[A]^{1}, …, [A]^{n}} 이 \mathbb{R}^{n} 의 orthonormal basis 이면, linear map L{A} 는 rigid motion 이다.

(관찰 9.2.10) L(0) = 0 인 \mathbb{R}^{n} 의 rigid motion L 이 \mathbb{R}^{n} 의 orthonormal basis \mathfrak{B} = {X_{i}} 를 고정하면 – 즉 , [L(X_{j}) = X_{j} for all j = 1, …, n ] 이면, - L은 항등사상이다.

(정리 9.2.11) L(0) = 0 인 \mathbb{R}^{n} 의 rigid motion L 은 linear map 이다.

따라서, \mathbb{R}^{n} 의 rigid motion M 은 translation 과 linear rigid motion 의 합성으로 쓸 수 있다.

(따름정리 9.2.12) \mathbb{R}^{n} 의 rigid motion 은 항상 bijection 이다.

(정의 9.2.13) \mathbb{R}^{n} 의 linear rigid motion 을 (real) orthogonal operator 라고 부른다. 또, L 이 \mathbb{R}^{n} 의 orthogonal operator 이고 L = L_{A} 일 때, real (n \times n)-matrix A 를 (real) orthogonal matrix 라고 부른다.

(subsetion 9.3.) Orthogonal operator / Matrix

(정의 9.3.1.) \mathbb{R}^{n} 의 orthogonal operator 전체의 집합을 [orthogonal group on \mathbb{R}^{n}] 이라고 부르고, \mathbf{O}(\mathbb{R}^{n}) 이라고 부른다. 즉, \mathbf{O}(\mathbb{R}^{n}) = {L \in \mathfrak{L} (\mathbb{R}^{n}, \mathbb{R}^{n}) | \Vert L(X) – L(Y) \Vert = \Vert X – Y \Vert for all X, Y \in \mathbb{R}^{n}} .

또, real ( n \times n )-orthogonal matrix 전체의 집합을 (real) orthogonal group 이라고 부르고 \mathbf{O}(n) 으로 표기한다. 즉, \mathbf{O}(n) = {A \in \mathfrak{M}_{n,n} (\mathbb{R}) | \Vert AX – AY \Vert = \Vert X – Y \Vert for all X, Y \in \mathbb{R}^{n}} 이다.

(따름정리 9.3.2.) M : \mathbb{R}^{n} \in \mathbb{R}^{n} 이 rigid motion 이면, M(X) = AX + B (X \in \mathbb{R}^{n}) 인 A \in \mathbf{O}(n) 과 B \in \mathbb{R}^{n}이 유일하게 존재한다.

(관찰 9.3.3.) L:\mathbb{R}^{n} \to \mathbb{R}^{n} 이 linear 일 때, 다음 조건들은 동치이다.

\Vert LX – LY \Vert = \Vert X – Y \Vert for all X, Y \in \mathbb{R}^{n} . (즉, L \in \mathbf{O}(\mathbb{R}^{n}).)

\Vert LX \Vert = \Vert X \Vert for all X \in \mathbb{R}^{n}.

\langle LX, LX \rangle = \langle X, X \rangle for all X \in \mathbb{R}^{n}

\langle LX, LY \rangle = \langle X, Y \rangle for all X, Y \in \mathbb{R}^{n}.

\mathfrak{B} 가 \mathbb{R}^{n} 의 orthonormal basis 이면, L(\mathfrak{B}) 도 \mathbb{R}^{n} 의 orthonormal basis 이다.

(관찰 9.3.5.) A \in \mathfrak{M}_{n,n} (\mathbb{R}) 일 때, 다음 조건들은 동치이다.

\Vert AX – AY \Vert = \Vert X – Y \Vert for all X, Y \in \mathbb{R}^{n} . (즉, A \in \mathbf{O}(n).)

\Vert AX \Vert = \Vert X \Vert for all X \in \mathbb{R}^{n}.

\langle AX, AX \rangle = \langle X, X \rangle for all X \in \mathbb{R}^{n}

\langle AX, AY \rangle = \langle X, Y \rangle for all X, Y \in \mathbb{R}^{n}.

A의 column 들의 집합 {[A]^{1}, …, [A]^{n}} 은 \mathbb{R}^{n} 의 orthonormal basis.

^{t}A \cdot A = I = A \cdot ^{t}A . 즉, A^{-1} = ^{t}A.

^{t}A \in \mathbf{O}(n).

A의 row들의 집합 {[A]{1}, …, [A]{n}} 은 \mathfrak{M}_{1,n}(\mathbb{R}) 의 orthonormal basis.

(정의 9.3.8) Special orthogonal group \mathbf{SO}(n) 을 \mathbf{SO}(n) = {A \in \mathbf{O}(n) | det(A) = 1} 로 정의한다. 이제, \mathbf{SO}(\mathbb{R}^{n}) = {L \in \mathbf{O}(\mathbb{R}^{n}) |det(L) = 1} 로 표기한다.

기하학에서 A \in \mathbf{SO}(n) 이면 L_{A} 를 \mathbb{R}^{n} 의 orientation preserving orthogonal operator, A \in \mathbf{O}(n) - \mathbf{SO}(n) 이면 L_{A} 를 \mathbb{R}^{n} 의 orientation reversing orthogonal operator 라고 부른다.

(subsection 9.4.) Reflection

(정의 9.4.2.) 0 \neq Y \in \mathbb{R}^{n} 일 때, 함수 S_{Y} : \mathbb{R}^{n} \to \mathbb{R}^{n} 을 S_{Y} (X) = X - \frac{2 \langle X , Y \rangle}{\langle Y, Y \rangle} Y (X \in \mathbb{R}^{n}) 으로 정의하고, S_{Y} 를 Y에 관한 reflection 또는 symmetry 라고 부른다.

(subsection 9.5.) \mathbf{O}(2) 와 \mathbf{SO}(2)

(정리 9.5.2.) \mathbf{SO}(2) 의 원소는 2-dimensional rotation 뿐이다. 즉, \mathbf{SO}(2) = {\begin{pmatrix}cos \theta & -sin \theta \ sin \theta & cos \theta \end{pmatrix} | 0 \le \theta < 2\pi} = {\begin{pmatrix} x & -y \ y & x \end{pmatrix} |x,y \in \mathbb{R} , x^{2} + y^{2} = 1}

(따름정리 9.5.3.) \mathbf{O}(2) 의 구조는 다음과 같다. 즉, \mathbf{O}(2) = \mathbf{SO}(2) \mathfrak{U} {\begin{pmatrix} x & -y \ y & x \end{pmatrix} |x,y \in \mathbb{R} , x^{2} + y^{2} = 1}

(관찰 9.5.6.) \theta \in \mathbb{R} 이면, S_{\theta} = S_{^{t}(-sin \frac{\theta}{2} , cos \frac{\theta}{2} } 이 성립한다.

따라서, \mathbb{R}^{2} 의 모든 reflection 은 S_{\theta} 의 꼴이다.

(따름정리 9.5.7.)

\mathbb{R}^{2} 의 orthogonal operator L 은 rotation 이거나 reflection 둘 중 하나이다. 즉, det(L) = 1 이면 L 은 rotation 이고, det(L) = -1 이면 L 은 reflection 이다.

따라서, \mathbb{R}^{2} 의 rotation \mathbb{R} 과 reflection S 의 합성 R \bullet S 와 S \bullet R 은 언제나 \mathbb{R}^{2} 의 reflection 이다.

역으로, \mathbb{R}^{2} 의 reflection S를 하나 고정하면, \mathbb{R}^{2} 의 모든 reflection 은 [S \bullet R for some rotation R] 의 꼴로 유일하게 쓸 수 있다.

(관찰 9.5.8.) S가 \mathbb{R}^{2} 의 reflection 이고 \theta \in \mathbb{R} 이면, S^{-1} \bullet R_{\theta} \bullet S = (R_{\theta})^{-1} , 즉 R_{\theta} \bullet S = S \bullet R_{-\theta} 가 성립한다.

(subsection 9.6.) \mathbf{SO}(3) 와 \mathbf{SO}(n)

(정의 9.6.1.) Z \in \mathbb{R}^{3} 가 unit vector 일 때, \mathfrak{B} = {Z,X,Y} 가 \mathbb{R}^{3} 의 rotation basis 인 X,Y 를 찾자. 그리고, det(Z,X,Y) = det(X,Y,Z) = 1 이 되도록 X,Y 의 순서를 정하자. 이 때, 원점을 중심으로 하는 \mathbb{R}^{3} 의 rotation R_{Z,\theta} : \mathbb{R}^{3} \to \mathbb{R}^{3} 를

[R_{Z,\theta}]_{\mathfrak{B}}^{\mathfrak{B}} = \begin{pmatrix} 1 & 0 & 0 \ 0 & cos \theta & -sin \theta \ 0 & sin \theta & cos \theta \end{pmatrix}

라고 정의하자. (이 때, Z를 회전축, \langle X \rangle \oplus \langle Y \rangle 을 회전판, \theta 를 회전각 이라고 부르면 자연스럽다.)

(정리 9.6.3.) \mathbf{SO}(\mathbb{R}^{3}) 의 원소는 모두 \mathbb{R}^{3} 의 rotation 이다.

(따름정리 9.6.4.) \mathbb{R}^{3} 의 두 rotation 의 합성도 rotation 이다.

(정의 9.6.6.) 임의의 n 에 대하여, \mathbf{SO}(\mathbb{R}^{n}) 의 원소를 – 혹은, \mathbf{SO} (n)의 원소를 - \mathbb{R}^{n} 의 rotation 이라고 정의한다.

Inner Product Space

(Section 10) 내적공간

(subsection 10.1) Inner product space

(정의 10.1.1) X = ^{t}(a_{1}, …, a_{n}) , Y = ^{t}(b_{1}, …, b_{n}) \in \mathbb{C}^{n} 일 때, \langle X , Y \rangle = ^{t} X \cdot \bar{Y} = \sum_{i =1}^{n} a_{i} \bar{b_{i}} 로 정의하고 이를 \mathbb{C}^{n} 의 (Hermitian) dot product 라고 부른다.

(정의 10.1.4) (F = \mathbb{R} 혹은 F = \mathbb{C} 일 때, ) F^{n} \times F^{n} 에서 F로 가는 함수 (X,Y) \mapsto \langle X, Y \rangle 가, 모든 X,Y,Z \in F^{n}, c \in F 에 대해, 다음 조건

\langle X + Y , Z \rangle = \langle X, Z \rangle + \langle Y , Z \rangle

\langle cX , Y \rangle = c \langle X , Y \rangle

\langle X, Y \rangle = \bar{\langle Y , X \rangle}

X \neq 0 이면, (\langle X, X \rangle \in \mathbb{R} 이고) \langle X, X \rangle > 0

을 만족하면 , \langle , \rangle 을 F^{n} 의 inner product 라고 부른다. (특별히, F = \mathbb{C}인 경우에는 Hermitian inner product 라고 부르기도 한다.)

(관찰 10.1.5) \langle , \rangle 가 F^{n} 의 inner product 라면 다음이 성립한다.

\langle X, Y + Z \rangle = \langle X , Y \rangle + \langle X , Z \rangle

\langle X , cY \rangle = \bar{c} \langle X, Y \rangle

(정의 10.1.6) (F = \mathbb{R} 혹은 F = \mathbb{C} 일 때, ) V를 F-vector space 라고 하자. (V가무한차원인 경우도 허용한다.) 이 때, 함수 \langle, \rangle : V \times V \to F 가, 모든 u,v,w \in V, c \in F 에 대해, 다음 조건

\langle u + v , w \rangle = \langle u, w \rangle + \langle v , w \rangle

\langle cv , w \rangle = c \langle v , w \rangle

\langle v, w \rangle = \bar{\langle w , v \rangle}

v \neq 0 이면, (\langle v, v \rangle \in \mathbb{R} 이고) \langle v, v \rangle > 0

을 만족하면 , \langle , \rangle 을 V 의 inner product 라고 부르고, inner product 가 주어진 V를inner product space (내적공간) 라고 부른다. (특별히, F = \mathbb{C} 인 경우에는 Hermitialn inner product 라고 부르기도 한다.)

(subsection 10.2) Inner product space 의 성질

(정의 10.2.2) v \in V 일 때, \Vert v \Vert = \sqrt{\langle v, v \rangle} 로 표기하고, \Vert v \Vert 를 v의 norm 이라고 부른다. Norm은 vector 의 길이(크기)로 이해한다.

(정의 10.2.5) v, w \in V 일 때, \langle v, w \rangle = 0 이면, v \perp w로 표기하고 v와 w는 서로 수직(perpendicular 또는 orthogonal) 이라고 말한다. (\langle v, w \rangle = 0 이면, \langle w ,v \rangle = 0 이다.)

S,T \subseteq V 일 때, [\langle v,w \rangle = 0 for all v \in S, w \in T ] 이면, S \perp T 로 표기하고 S 와 T는 서로 수직이라고 말한다.

(관찰 10.2.8) v,w \in V 이면, 다음이 성립한다.

(Cauchy-Schwarz Inequality) | \langle v, w \rangle | \le \Vert v \Vert \cdot \Vert w \Vert (등호가 성립할 필요충분조건은 {v,w} 가 일차종속인 것이다.)

(Triangle inequality) \Vert v + w \Vert \le \Vert v \Vert + \Vert w \Vert

(정의 10.2.10) V의 non-zero vector 들 {v_{i} | i \in I} 가 mutually perpendicular 이면 – 즉 [v_{i} \perp v_{j} for all i \neq j \in I] 이면 – {v_{i} | i \in I} 를 V의 orthogonal subset 이라고 부른다. 이 때, v_{i} 들이 모두 unit vector 이면, 이번에도 {v_{i} | i \in I } 를 V의 orthonormal subset 이라고 부른다.

V의 basis \mathfrak{B} 가 orthogonal subset 이면, \mathfrak{B} 를 V의 orthogonal basis 라고 부르고, orthonormals subset 이면 \mathfrak{B} 를 V의 orthonormal basis 라고 부른다.

(정의 10.2.14) S \subseteq V 일 때, S^{\perp} = {v \in V | v \perp w for all w \in S} 로 표기한다. 특히, W \le V 일 때에는 W^{\perp} 를 W의 orthogonal complement 라고 부른다.

(subsection 10.3) Gram-Schmidt Orthogonalization

(Gram-Schmidt Orthogonalization) (정리 10.3.1) V가 inner product space 이고 {v_{1}, …, v_{r}} 가 V의 linearly independent subset 이라고 하자.

w_{1} = v_{1} 으로, 그리고 2 \le i \le r 일 때에는 w_{i} = v_{i} - \frac{\langle v_{i}, w_{i-1} \rangle}{\langle w_{i-1} , w_{i-1} \rangle} w_{i-1} - \cdots - \frac{\langle v_{i}, w_{2} \rangle}{\langle w_{2}, w_{2} \rangle} w_{2} - \frac{\langle v_{i}, w_{1} \rangle}{ \langle w_{1}, w_{1} \rangle} w_{1} 으로 inductively 정의하면, \langle v_{1}, …, v_{r} \rangle = \langle w_{1}, …, w_{r} \rangle 이고, {w_{1}, …, w_{r}} 은 V의 orthogonal subset 이 된다.

따라서, {\frac{1}{\Vert w_{1} \Vert} w_{1} , …, \frac{1}{\Vert w_{r} \Vert} w_{r}} 은 V의 orthonormal subset 이 된다.

특별히, r = dim V 이면 , {\frac{1}{\Vert w_{1} \Vert} w_{1} , …, \frac{1}{\Vert w_{r} \Vert} w_{r}} 은 V의 orthonormal basis 가 된다.

(따름정리 10.3.4.) V가 유한차원 inner product space 이고 W \le V 라고 하자.

W 자신 inner product space 이므로, W도 orthonormal basis 를 갖는다.

따라서, V = W \oplus W^{\perp} 이고, 특별히 dim V = dim W + dim W^{\perp} 이다.

(따름정리 10.3.7) A \in \mathfrak{M}{m,n}(\mathbb{R}) 의 row space \langle ^{t}[A]{1}, …, ^{t}[A]{m} \rangle 은 연립방정식 AX = 0 의 solution space 의 orthogonal complement 이다. 즉, ker (L{A}) = {X \in \mathbb{R}^{n} | AX = 0} = \langle ^{t}[A]{1}, …, ^{t}[A]{m} \rangle^{\perp} 이다. 따라서, (따름정리 10.3.4.)에 의해 dim ker(L_{A}) = n – [row rank of A] 이다.

(subsection 10.4.) Standard Basis 대 Orthonormal Basis

(관찰 10.4.1.) \mathfrak{B} = {v_{1}, …, v_{n}} 이 inner product space V 의 orthogonal basis 이면, 다음이 성립한다.

v \in V 이면, v = \sum_{i = 1}^{n} \frac{\langle v,v_{i} \rangle}{\langle v_{i}, v_{i} \rangle} v_{i} (즉, [v]{\mathfrak{B}} 의 i-번째 좌표는 \frac{\langle v, v{i} \rangle}{\langle v_{i}, v_{i} \rangle}).

그리고, \mathfrak{B} = {v_{1}, …, v_{n}} 이 inner product space V 의 orthonormal basis 이면, 다음이 성립한다.

v \in V 이면, v = \sum_{i = 1}^{n} \langle v,v_{i} \ranglev_{i} (즉, [v]{\mathfrak{B}} 의 i-번째 좌표는 \langle v, v{i} \rangle).

v,w \in V 이면, \langle v, w \rangle = ^{t} [v]{\mathfrak{B}} \cdot \bar{[w]{\mathfrak{B}}}.

(정의 10.4.2.) \mathfrak{B} = {v_{i} | i \in I} 가 inner product space V 의 orthonormal subset 이라고 하자. v \in V 일 때, \langle v, v_{i} \rangle 을 \mathfrak{B} 에 관한 v의 i-th Fourier coefficient 라고 부른다. 그리고, 이를 \mathfrak{B} 에 관한 v 의 i-번째 좌표로 생각한다.( 한편, \mathfrak{B} 가 V의 orthogonal subset 일 때에는 \frac{\langle v, v_{i}\rangle}{\langle v_{i}, v_{i} \rangle} 을 \mathfrak{B} 에 관한 v의 i-th Fourier coefficient 라고 부른다.)

(관찰 10.4.3.) W가 inner product space V의 finite dimensional subspace 이고, {v_{1}, …, v_{m}} 을 W의 orthonormal basis 라고 하자. v \in V 이면, v = w + w’ 인 w \in W 와 w’ \in W^{\perp} 가 유일하게 존재하고, w = \langle v, v_{1} \rangle v_{1} + \cdots + \langle v, v_{m} \rangle v_{m} 으로 주어진다.

(Closest Vector Problem) (관찰 10.4.4) 위 (관찰 10.4.3) 의 w는 v에 가장 가까운 W의 vector 이다.

(Bessel’s Inequality) (관찰 10.4.7) \mathfrak{B} = {v_{1}, …, v_{n}} 이 inner product space V 의 orthonormal subset 이면, 모든 v \in V 는 다음 부등식 \sum_{i = 1}^{n} |\langle v, v_{i} \rangle |^{2} \le \Vert v \Vert^{2} 을 만족한다.

(subsection 10.5) Inner product space 의 isomorphism

(정의 10.5.3) V 와 V’ 이 F-위의 inner product space 이고, 다음 조건 \langle \phi(v) , \phi(w) \rangle = \langle v, w \rangle , (v,w \in V) 을 만족하는 vector space isomorphism \phi : V \to V’ 이 존재하면, 우리는 V와 V’ 이 inner product space 로서 isomorphic 하다고 말하고, \phi 를 inner product space isomorphism 이라고 부른다.

(Classification of finite dimensional inner product space) (정리 10.5.5) V 와 W가 F-위의 유한차원 inner product space 일 때, 다음은 동치이다.

V와 W는 inner produce space 로서 isomorphic.

dim V = dim W.

(정리 10.5.6.) V가 \mathbb{R} -위의 유한차원 inner product space 일 때,

L(0) = 0 인 V의 rigid motion L 은 linear map 이다.

따라서, V 의 rigid motion M 은 translation 과 linear rigid motion 의 합성으로 쓸 수 있다.

V의 rigid motion 은 항상 bijection 이다.

(section 10.6) Orthogonal Group 과 Unitary Group

(정의 10.6.1) (F = \mathbb{R}) 인 경우) V가 inner product \langle , \rangle 가 주어진 \mathbb{R}-위 의 inner product space일 때, \mathbf{O}(V) = \mathbf{O}(V, \langle, \rangle ) = {L \in \mathfrak{L}(V,V) | \Vert Lv – Lw \Vert = \Vert v – w \Vert for all v, w \in V} 로 표기하고 [orthogonal group on V with respect to \langle , \rangle ] 라고 부른다. 또, \mathbf{O}(V) 의 원소를 orthogonal operator 라고 부른다.

(F = \mathbb{C}) 인 경우) V가 inner product \langle , \rangle 가 주어진 \mathbb{C}-위 의 inner product space일 때, \mathbf{U}(V) = \mathbf{U}(V, \langle, \rangle ) = {L \in \mathfrak{L}(V,V) | \Vert Lv – Lw \Vert = \Vert v – w \Vert for all v, w \in V} 로 표기하고 [unitary group on V with respect to \langle , \rangle ] 라고 부른다. 또, \mathbf{U}(V) 의 원소를 unitary operator 라고 부른다.

(정의 10.6.2.) \mathbb{C}^{n} 에 (Hermitian ) dot product 가 주어졌을 때, \mathbf{U}(n) = {A \in \mathfrak{M}{n,n}(\mathbb{C}) | L{A} \in \mathbf{U} (\mathbb{C}^{n} , dot product )} = {[L]{\mathcal{E}}^{\mathcal{E}} \in \mathfrak{M}{n,n}(\mathbb{C}) | L \in \mathbf{U} (\mathbb{C}^{n} , dot product )} 로 표기하고, 이를 (complex) unitary group 이라고 부른다. 그리고, \mathbf{U}(n) 의 원소를 (complex) unitary matrix 라고 부른다.

(정의 10.6.3.) V가 inner product \langle , \rangle 가 주어져 있는 F-위의 n-dimensional inner product space 이고, \mathfrak{B} 를 V의 orthonormal basis 라고 하자.

F = \mathbb{R} 이면, \mathbf{O}{n} (\mathbb{R} , \langle, \rangle ) = {[L]{\mathfrak{B}}^{\mathfrak{B}} \in \mathfrak{M}_{n,n}(\mathbb{R}) | L \in \mathbf{O} (V, \langle, \rangle ) } 로 표기하고, 이를 V와 \langle, \rangle 로부터 얻어진 orthogonal group 이라고 부른다.

F = \mathbb(C) 이면, \mathbf{U}{n} (\mathbb{C} , \langle, \rangle ) = {[L]{\mathfrak{B}}^{\mathfrak{B}} \in \mathfrak{M}_{n,n}(\mathbb{C}) | L \in \mathbf{U} (V, \langle, \rangle ) } 로 표기하고, 이를 V와 \langle, \rangle 로부터 얻어진 unitary group 이라고 부른다.

(명제 10.6.4) 위 (정의 10.6.3) 은 – 즉, \mathbf{O}{n} (\mathbb{R} , \langle, \rangle ) 및 \mathbf{U}{n} (\mathbb{C} , \langle, \rangle) 의 정의는 V의 orthonormal basis 의 선택과는 무관하다.

(따름명제 10.6.5.)

\mathfrak{B} 가 [dot product 가 주어진 Euclidean space \mathbb{R}^{n}] 의 임의의 orthonormal basis 이면, \mathbf{O}(n) = {[L]{\mathfrak{B}}^{\mathfrak{B}} \in \mathfrak{M}{n,n}(\mathbb{R}) | L \in \mathbf{O}(\mathbb{R}^{n})} 이다. (단, \mathbf{O}(\mathbb{R}^{n}) = \mathbf{O}(\mathbb{R}^{n}, dot product) .)

\mathfrak{B} 가 [(Hermitian) dot product 가 주어진 \mathbb{C}^{n}] 의 임의의 orthonormal basis 이면, \mathbf{U}(n) = {[L]{\mathfrak{B}}^{\mathfrak{B}} \in \mathfrak{M}{n,n}(\mathbb{C}) | L \in \mathbf{U}(\mathbb{C}^{n})} 이다. (단, \mathbf{U}(\mathbb{C}^{n}) = \mathbf{U}(\mathbb{C}^{n}, dot product) .)

(관찰 10.6.6.) V가 inner product \langle , \rangle 가 주어진 n-dimensional inner product space 이고, L \in \mathfrak{L} (V,V) 일 때, 다음 조건들은 동치이다.

\Vert Lv – Lw \Vert = \Vert v – w \Vert for all v, w \in V.

\Vert Lv \Vert = \Vert v \Vert for all v \in V.

\langle Lv, Lv \rangle = \langle v, v \rangle for all v \in V.

\langle Lv, Lw \rangle = \langle v, w \rangle for all v, w \in V.

\mathfrak{B} 가 V의 orthonormal basis 이면, L(\mathfrak{B}) 도 V의 orthonormal basis.

(정의 10.6.9) A \in \mathfrak{M}{m,n}(\mathbb{C}) 일 때,A^{} = \bar{^{t}A} = ^{t}\bar{A} 로 표기하고, A^{} 를 A의 adjoint matrix 라고 부른다. (A \in \mathfrak{M}{n,n}(\mathbb{R}) 일 때에는, 물론 A^{*} = ^{t} A로 이해한다.)

(정의 10.6.13.) 다음 표기법 \mathbf{SU} (n) = {A \in \mathbf{U}(n) | det(A) = 1} 이다. \mathbf{SU} (n) 을 special unitary group 이라고 부른다.

(subsection 10.7.) Adjoint Matrix 와 그 응용

(관찰 10.7.1.) F^{n} 과 F^{m} 에 각각 dot product \langle , \rangle 가 주어져 있다고 하자. 이 때, A \in \mathfrak{M}_{n,n} (F) 이면, \langle AX, Y \rangle = \langle X, A^{} Y \rangle , \langle Y , AX \rangle = \langle A^{} Y , X \rangle , (X \in F^{n}, Y \in F^{m}) 이 성립한다.

(정의) AX = B 가 solution 을 갖지 않는 경우에 \Vert AX_{0} – B \Vert 가 최소인 X_{0} 을 구하는 문제를 최소자승법 (Least Squares Approximation – LSA) 라고 한다.

(명제 10.7.3.) A \in \mathfrak{M}_{m,n} (F) 이고 B \in F^{m} 일 때,

LSA-문제의 solution – 즉, [\Vert AX_{0} – B \Vert \le \Vert AX – B \Vert for all X in F^{n} ] 인 X_{0} \in F^{n} 은 항상 존재한다.

만약 L_{A} 가 injective 이면, LSA-문제의 solution X_{0} 는 유일하게 결정된다.

(명제 10.7.4.) A \in \mathfrak{M}_{m,n}(F) 이고 B \in F^{m} 일 때, 다음은 동치이다.

X_{0} 는 LSA-문제의 solution (즉, \Vert AX_{0} – B \Vert \le \Vert AX – B \Vert for all X \in F^{n} )

(A^{} A ) X_{0} = A^{} B.

(명제 10.7.5) A \in \mathfrak{M}{m,n} (F) 이고 B \in F^{m} 이라고 하자. 만약, m \ge n 이고 rk(A) = n 이면 (즉 A가 full rank 를 가지면), LSA-문제의 solution X{0} 은 존재하고 하나뿐이다.

(관찰 10.7.9.) A \in \mathfrak{M}_{m,n}(F) 이면, rk(A^{} A) = rk(A) 이다. (따라서, A가 full rank 를 가지면, (A^{}A) 는 가역행렬이다.)

(관찰 10.7.11) A \in \mathfrak{M}_{m,n}(F) 이면, 다음이 성립한다.

im(A^{}) = (ker A)^{\perp} . 즉, ker A = (im(A^{})^{\perp}

im(A) = (ker(A^{}))^{\perp}. 즉, ker(A^{}) = (im A)^{\perp}

(정의) Minimal solution problem 은 AX = B가 solution 을 가질 때, 크기가 가장 작은 solution 을 구하는 것이다.

(명제 10.7.12.) A \in \mathfrak{M}_{m,n} (F) 이고 B \in F^{m} 일 때, AX = B 가 solution 을 갖는다고 가정하자. 그러면

AX = B 의 minimal solution – 즉 , AX_{0} = B 이고 [X \in F^{n} 이 AX = B 를 만족하면 \Vert X_{0} \Vert \le \Vert X \vert ] 인 X_{0} \in F^{n} 이 유일하게 존재한다.

{X_{0}} = [AX = B 의 해집합] \cap im(A^{*}) .

Group

(Section 11) 군

(subsection 11.1) Binary Operation 과 Group

(정의 11.1.1.) 집합 X가 있을 때, 함수 * : X \times X \to X 를 X위의 이항연산 (Binary operation) 이라고 부른다.

(정의 11.1.2.) 이항연산을 갖는 집합 X가 있을 때, 임의의 x,y,z \in X 에 대해 (xy)z = x(yz) 가 성립하면, 이항연산이 결합법칙(Associative law) 를 만족한다고 말한다.

(정의 11.1.4.) 이항연산 *를 갖는 집합 G가 다음 조건들

모든 x,y,z \in G 에 대하여 (xy)z = x(yz)

[모든 x \in G 에 대하여 xe = ex = x] 인 원소 e \in G 가 존재.

각 x \in G 에 대하여 x\bar{x} = \bar{x} x = e 를 만족하는 원소 \bar{x} \in G 가 존재.

를 만족하면, G를 , 정확하는 (G,*) 를 , 군 (Group) 이라고 부른다.

(관찰 11.1.6.) 군은 공집합이 아니다.

(관찰 11.1.7.) G가 group 이면 [[모든 x \in G 에 대하여 xe = ex = x] 인 원소 e \in G 가 존재.] 조건을 만족하는 e는 하나뿐이다. 이 때 e를 G의 항등원( identity 혹은 identity element) 라 부르고, G를 강조할 필요가 있으면 e = e_{G} 로 표기한다.

(관찰 11.1.8.) G가 군이고 x \in G 이면, [각 x \in G 에 대하여 x\bar{x} = \bar{x} x = e 를 만족하는 원소 \bar{x} \in G 가 존재.] 를 만족하는 \bar{x}는 하나뿐이다. 이 때, \bar{x} 를 x의 역원 ( inverse element) 라고 부르고, \bar{x} = x^{-1} 로 표기한다.

(정의 11.1.9.) 군 G가 finite set 이면, G를 유한군(finite group) 이라고 부른다. 따라서, 무한군(infinite group) 의 뜻도 자명하다. 특히, G가 유한군일 때, |G| 를 G 의 order 라고 부른다.

(정의 11.1.10.) 군 G가 모든 x,y \in G 에 대해 xy = yx 의 조건을 만족하면, G를 가환군 (Commutative group, 혹은 abelian group) 이라고 부른다.

(subsection 11.2.) Group 의 초보적인 성질

(표기법 11.2.1) (Multiplicative notation) Group G 의 이항연산을 곱셈으로 표기할 때는 항등원을 대개 e = 1로 표기한다. 그리고 g \in G 일 때, 언제나처럼 g^{1} = g, g^{2} = gg , \cdots 로 정의한다. g^{0} =1 로 정의하는 것이 관습이고, g^{-2} = g^{-1}g^{-1}, g^{-3} = g^{-1}g^{-1}g^{-1} , \cdots 로 표기한다.

(Additive Notation) Group G의 이항연산을 덧셈으로 표기할 때에는 보통 G가 abelian group 인 것을 implicitly 가정한다. 이 때에는, 항등원을 e = 0 으로 표기하고 g \in G 의 inverse element 를 (-g) 로 표기하는 것이 관습이다. 따라서 이 경우에는 \cdots , (-2)g = (-g) + (-g) , (-1)g = -g , 0g = 0, 1g = g, 2g = g+g , \cdots 의 표기법이 자연스럽다. (그리고 h \in G 일 때, g-h = g + (-h) 의 표기법도 사용한다.)

G가 commutative group 이라고 하면 multiplicative notation을 사용하고, abelian group 이라고 하면 additive notation 을 사용하는 것이 관례이다. 별다른 언급이 없으면 multiplicative notation 을 사용한다.

(Cancellation Law) (관찰 11.2.9.) x,y,z \in G 일 때, xy = xz 이면 y = z 이다. 또, xz = yz 이면 x = y 이다.

(관찰 11.2.14.) 연산표의 각 가로줄에는 모두 다른 원소가 나타난다. 각 세로줄도 마찬가지이다.

(관찰 11.2.15.) 고정된 x \in G 에 대해서 함수 \lambda_{x} : G \to G 를 \lambda_{x} (y) = xy ( y \in G) 로 정의하면, \lambda_{x} : G \to G 는 bijection 이다. \lambda_{x} 는 집합 G를 permute 한다.

(subsection 11.3.) Subgroup

(정의 11.3.1.) H가 group G의 subset일 때( 즉 H \subseteq G일 때) G로부터 물려받은 binary operation 에 관하여, H 자신 group 이 되면, 우리는 H를 G의 subgroup(부분군) 이라고 부르고 H \le G 로 표기한다.

(관찰 11.3.2.) H \le G 일 때,

G 의 identity element 를 e_{G} 로, H 의 identity element 를 e_{H} 로 표기하면, e_{H} = e_{G} 이다.

h \in H 일 때, h 의 G에서의 inverse 를 h^{-1} 로, H에서의 inverse 를 h’ 으로 표기하면, h’ = h^{-1} 이다.

(관찰 11.3.4.) Group G 의 subset H 가 G의 subgroup 일 필요충분조건은 다음과 같다.

h_{1} , h_{2} \in H 이면, h_{1}h_{2} \in H .

e \in H.

h \in H 이면, h^{-1} \in H.

(관찰 11.3.7.) K \le H 이고 H \le G 이면, K \le G 이다. (즉, 부분군의 부분군은 부분군이다.)

(표기법 11.3.9.) H , K \subseteq G 일 때, HK = {hk \in G | h \in H, k \in K}로 표기한다.

(정의 11.3.11.) S \subseteq G 일 때, S를 포함하는 가장 작은 G의 subgroup 을 \langle S \rangle 로 표기하고, \langle S \rangle 을 subgroup generated by S (S 가 생성한 부분군) 이라고 부른다. (여기서 ‘가장 작다’는 말은 [물론 S \subseteq \langle S \rangle \le V 이면서, 만약 H 가 S를 포함하는 임의의 G 의 subgroup 이면 \langle S \rangle \le H ] 라는 뜻이다.)

(관찰 11.3.12.) S \subseteq G 이면, \langle S \rangle = \bigcup_{S \subseteq H \le G} H 이다. 이로부터 \langle S \rangle 의 existence 와 uniqueness 가 얻어진다.

(정의 11.3.15.) x \in G 일 때, \langle x \rangle 를 x 가 generate 하는 G의 cyclic subgroup 이라고 부른다.

G = \langle x \rangle 인 x \in G 가 존재하면, G 를 x 를 generator 로 갖는 cyclic group 이라고 부른다.

(정의 11.3.16.) S \subseteq G 일 때, S^{-1} = {s^{-1} \in G | s \in S } 로 표기하자. 이 때, S \cup S^{-1} 의 원소를 alphabet 이라고 부르자.

Alphabet 들을 유한개 늘어놓은 (곱한) 것을 word 라고 부른다. e는 empty word 로 취급한다.

(관찰 11.3.17) S \subseteq G 일 때, [alphabet S \cap S^{-1} 로 만들어진 word 전체의 집합] 은 G의 subgroup 이 된다.

(명제 11.3.18) S \subseteq G 이면, \langle S \rangle 는 [alphabet S\cupS^{-1} 로 만들어진 word 전체의 집합]과 같다.

(subsection 11.4.) 학부 대수학의 반

(학부 대수학의 반) (정리 11.4.2.) Abelian group (\mathbb{Z}, +) 의 subgroup 은 n\mathbb{Z} 들 뿐이다. (단, 0 \le n \in \mathbb{Z})

(subsection 11.5.) Group Isomorphism

(정의 11.5.1.) G와 G’이 group 이고, 다음 조건 \phi(g_{1}g_{2}) = \phi(g_{1}) \phi(g_{2}) (g_{1}, g_{2} \in G) 를 만족하는 bijection \phi : G \to G’ 이 존재하면, 우리는 group G와 G’ 이 (group 으로서) isomorphic 하다고 말하고, G \approx G’ 혹은

\begin{tikzpicture}\matrix (m) [matrix of math nodes,row sep=3em,column sep=4em,minimum width=2em] {G &G’ \}; \path[-stealth] (m-1-1) edge node [above] {$\phi$} node [below]{$\approx$} (m-1-2);\end{tikzpicture}

으로 표기한다. 이 때, \phi 를 (grou) isomorphism(from G onto G’) 이라고 부른다.

(관찰 11.5.4.) 위 (정의 11.5.1.) 의 relation \approx 는 equivalence relation 이다.

(subsection 11.6.) Group Homomorphism

(정의 11.6.1.) G,H 가 group 일 때, 함수 \phi : G \to H 가 다음 조건 \phi(g_{1}g_{2}) = \phi(g_{1}) \phi(g_{2}) (g_{1}, g_{2} \in G) 를 만족하면 \phi 를 (group) homomorphism from G into H 라고 부른다.

(관찰 11.6.2.) \phi : G \to H 가 group homomorphism 이면,

\phi(e) = e

x \in G 이면, \phi(x^{-1}) = (\phi(x))^{-1} = \phi x^{-1}.

(정의 11.6.3.) \phi : G \to H 가 homomorphism 일 때,

\phi 가 injective 이면, \phi 를 monomorphism 이라고 부른다.

\phi 가 surjective 이면, \phi 를 epimorphism 이라고 부른다.

\phi 가 bijective 이면, \phi 를 isomorphism 이라고 부른다.

G = H 이면, \phi 를 endomorphism 이라고 부른다.

Bijective endomorphism 을 automorphism 이라고 부른다.

(관찰 11.6.4.) \phi : G \to H 가 homomorphism 일 때, 다음 조건은 동치이다.

\phi 는 isomorphism.

[\psi \bullet \phi = I_{G} 이고 \phi \bullet \psi = I_{H}] 인 homomorphism \psi : H \to G 가 존재한다.

(정의 11.6.5.) \phi : G \to H 가 homomorphism 일 때,

ker \phi = \phi^{-1} (e) = {x \in G | \phi(x) = e} 를 \phi 의 kernel 이라고 부른다.

im \phi = \phi(G) = {\phi(x) | x \in G } 를 \phi 의 image 라고 부른다.

(관찰 11.6.6.) \phi : G \to H 가 homomorphism 이면,

ker \phi \le G 이고 im \phi \le H 이다.

\phi 가 monomorphism 이면, G 와 im \phi 는 isomorphic 하다.

K \le G 이면, \phi(K) \le H 이다.

(관찰 11.6.7.) \phi : G \to H 가 homomorphism 일 때, 다음 조건은 동치이다.

\phi 는 monomorphism 이다.

x , y \in G 이고 \phi x = \phi y 이면, x = y 이다.

x \in G 이고 \phi x = \phi e = e 이면, x = e 이다.

ker \phi = {e} 이다.

(관찰 11.6.8.) \phi : G \to H 가 homomorphism 이고 S 가 G 의 subset 이면,

\phi \langle S \rangle = \lange \phi S \rangle 이다.

G = \langle S \rangle 이면, im \phi = \langle \phi S \rangle이다.

\phi 가 epimorphism 일 필요충분조건은 \phi 가 (G의) generating set 을 (H의) generating set 으로 옮기는 것이다.

(subsection 11.7.)

(정의 11.7.3.) x \in G 일 때, G의 cyclic subgroup \langle x \rangle 의 order 를 x 의 order 라고 부른다. x 의 order 는 |x| 로 표기한다. 즉 |\langle x \rangle | = |x|.

(관찰 11.7.4.) x \in G 일 때, |x| = n < \infty 일 필요충분조건은 다음과 같다.

x^{n} = 1 이고, 0 m < n 이면 x^{m} \neq 1. 즉, n 은 x^{n} = 1인 최소의 자연수이다.

(재정의 11.7.5.) x \in G 일 때, homomorphism \gamma_{x} : \mathbb{Z} \to G 를 \gamma_{x} (a) = x^{a}, (a \in \mathbb{Z}) 라고 정의하자. 이 때 ker \gamma_{x} \le \mathbb{Z} 이므로, [학부대수학의 반] 에 의해 ker \gamma_{x} = n\mathbb{Z} 인 0 \le n \in \mathbb{Z} 가 존재한다. 만약 n = 0 이면 |x| = \infty 로 정의하고, n \neq 0 이면 |x| = n 으로 정의한다.

(정리 11.7.8.) Cyclic group 은 up to isomorphism \mathbb{Z} 와 \mu_{n} 뿐이다.

(정리 11.7.9.) Cyclic group 의 subgroup 은 cyclic group 이다.

(subsection 11.8.) Group 과 Homomorphism 의 보기

(재정의 11.8.6.) X가 non-empty set 일 때, S_{X} 를 S_{X} = {\sigma : X \to X | \sigma 는 bijection} 으로 정의하면, (S_{X} , \bullet ) 는 group 이 된다. S_{X} 를 symmetric group on X 라고 부른다.

(Cayley’s Theorem) (정리 11.8.8.) \lambda : G \to S_{G} 를 (\lambda(x))(g) = \lambda_{x} (g) = xg (x,g \in G) 로 정의하면, \lambda 는 monomorphism 이 된다. 그러므로 모든 group 은 symmetric group 의 subgroup 과 isomorphic 하다.

(subsection 11.9.) Linear group

(보기 11.9.1.) 가역행렬 전체의 집합을 GL_{n}(F) = {A \in \mathfrak{M}{n,n}(F) | A is invertible} 으로 표기하고 [general linear group over F] 라고 부른다. GL{n}(F) 는 행렬의 곱셈을 이항연산으로 갖는 group 이다. 그리고, GL_{n}(F) 의 subgroup 을 linear group 또는 matrix group 으로 부른다.

SL_{n}(F) = {A \in GL_{n}(F) |det(A) = 1} 로 표기하고, [special linear group over F] 라고 부른다. SL_{n} (F) \le GL_{n}(F) 이다.

(보기 11.9.2.) V가 finite dimensional F-vector space 일 때, GL(V) = {L \in \mathfrak{L}(V,V) | L is invertible} 으로 표기하고 [general linear group on V] 라고 부른다. GL(V) 는 선형사상의 합성을 이항연산으로 갖는 group 이다. 그리고, GL(V) 의 subgroup 을 linear group 으로 부른다.

SL(V)= {L \in GL(V) |det(L) = 1} 로 표기하고, [special linear group on V] 라고 부른다. SL(V)\le GL(V) 이다.

(표기법 11.9.7.) GL_{1}(F), 즉 F – {0} 을 F^{\times} = GL_{1}(F) = F-{0} 으로 표기하고, [the multiplicative subgroup of F] 라고 부른다.

(Cayley’s Theorem 의 따름정리)(따름정리 11.9.21.) 모든 finite group 은 linear group (또는 matrix group) 과 isomorphic 하다.

Quotient

(Section 12) Quotient

(subsection 12.1) Equivalence class 와 Partition

(정의 12.1.1.) 집합 X가 있을 때, X \times X 의 subset R이 다음 조건을 만족하면, R 을 X-위의 equivalence relation 이라고 부른다. 또, (x,y) \in R 이면, x~_{R} y, 혹은 간단히 x ~ y 로 표기한다.

(x,x) \in R for all x \in X. (Reflexivity)

(x,y) \in R 이면, (y,x) \in R (Symmetry)

(x,y) \in R 이고 (y,z) \in R 이면 (x,z) \in R. (Transitivity)

(정의 12.1.2) ~ 이 집합 X-위의 equivalence relation 이고 x \in X 일 때, [x] = { y \in X | x ~ y} 로 표기하고, [x] 를 x의 (x를 대표로 갖는) equivalence class 라고 부른다. 또, X/~ = {[x] | x \in X} 로 표기하고, X/~ 를 the set of equivalence classes in X 라고 부른다.

(정의 12.1.5.) ~ 이 집합 X-위의 equivalence relation 일 때, X 의 subset \mathcal{C} 가 다음 조건을 만족하면, \mathcal{C} 를 a complete set of representatives 라고 부른다.

z \in X 이면, z ~ x for some x \in \mathcal{C}.

x,y \in \mathcal{C} 이면, x \nsim y.

이 때 X = \mathfrak{U}_{x \in \mathcal{C}} [x] 이다. 이를 equivalence class decomposition 이라고 부른다.

(정의 12.1.6.) X가 집합이고 A가 index set 일 때, X의 non-empty subset 들 {X_{\alpha} | \alpha \in A} 가 다음 조건

\bigcul_{\alpha \in A} X_{\alpha} = X.

X_{\alpha} \cap X_{\beta} = \empty if \alpha \neq \beta.

을 만족하면, {X_{\alpha} | \alpha \in A} 를 X-의 partition 이라고 부른다. 그리고, 각각의 X_{\alpha} 를 part 라고 부른다.

(관찰 12.1.7.) 집합 X위에 equivalence relation 이 주어졌다는 말과 partition 이 주어졌다는 말은 같은 말이다.

(표기법 12.1.12) H \le G 이고 g \in G 일 때, 다음 표기법 g^{-1} H g = {g^{-1} hg | h \in H} , gHg^{-1} = {ghg^{-1} | h \in H} 를 사용한다.

(subsection 12.2.) Coset

(정의 12.2.1.) H \le G 이고 x \in G 일 때, xH = {xh | h \in H} 로 표기하고 (G의 연산이, 예를들어 덧셈일 때에는 x + H 로 표기) xH 를 [x를 대표로 갖는 left coset modulo H] 라고 부른다. 마찬가지로, Hx = {hx | h \in H} 로 표기하고, Hx 를 [x를 대표로 갖는 right coset modulo H] 라고 부른다.

(관찰 12.2.2.) H \le G 이고 x,y \in G 일 때,

x \in xH 이고, 따라서 G = \bigcup_{x \in G} xH.

xH = yH iff y^{-1}x \in H.

xH \cap yH = \empty if xH \neq yH.

(표기법 12.2.4.) H \le G 이고 x,y \in G 일 때, 다음 표기법 x \equiv y (mod H) \iff xH = yH \iff x ~{H} y 와 \bar{x} = [x] = xH 를 사용한다. coset들 전체의 집합 – 즉 equivalence class 전체의 집합- 을 G/H = \frac{G}{H} = {xH | x \in G} = G/~{H} 로 표기하고, G/H 를 [coset space of G modulo H] 라고 부른다. 따라서 {x_{i} | i \in I} 를 a complete set of representatives 라고 하면, G/H = {\bar{x_{i}} | i \in I} 가 된다. 이 때, 서로 다른 coset 들 전체의 집합은 G의 partition 이므로, G = \mathfrak{U}{i \in I} x{i}H 로 쓸 수 있고, 이를 [coset decomposition of G modulo H] 라고 부른다.

(관찰 12.2.11.) H \le G 이고 x,y \in G 이면, |xH| = |yH| = |H| 이다.

(Lagrange’s Theorem) (관찰 12.2.12) G가 finite group 이고 H \le G 이면, |H| 는 |G| 의 약수이고, |G/H| = \frac{|G|}{|H|} 이다.

(표기법 12.2.13.) H \le G 일 때, |G/H| < \infty 이면, [G : H] = |G/H| 로 표기하고, [G:H] 를 [index of H in G]라고 부른다.

(관찰 12.2.16.) G가 finite group 일 떄 x \in G 이면, |x| \mid |G| 이다. 즉 x의 order 는 G의 order 를 나눈다.

(명제 12.2.17.) p가 prime 이고 |G| = p 이면, G 는 cyclic 이다. 즉, G \approx \mu_{p} 이다.

(Cauchy’s Theorem) (정리 12.2.22) p 가 prime 이고 p \mid |G| 이면, |x| = p 인 x \in G 가 존재한다.

(subsection 12.3.) Normal Subgroup 과 Quotient Group

(정의 12.3.2.) H \le G 일 때, 다음 조건 g^{-1}hg \in G (for all g \in G, h \in H) 를 만족하면, H 를 G의 normal subgroup 이라고 부르고, H \trianglelefteq G 로 표기한다.

(관찰 12.3.3.) H \le G 일 때, coset space G/H 의 binary operation \bar{x} \cdot \bar{y} = \bar{xy} (x,y \in G) 가 well-defined 되어있을 필요충분조건은 H \trianglelefteq G 로 표기한다.

(명제 12.3.4.) N \trianglelefteq G 일 때, coset space G/N 의 binary operation 을 bar{x} \cdot \bar{y} = \bar{xy} (x,y \in G) 라 정의하면, G/N 은 group 이 된다. 우리는 G/N 을 [quotient group of G modulo N] 이라고 부른다.

(관찰 12.3.6.) H \le G 일 때, 다음 조건은 동치이다.

H \trianglelefteq G.

g^{-1}Hg \le H for all g \in G.

g^{-1}Hg = H for all g \in G.

(관찰 12.3.10.) H \le G 이고 [G:H] = 2이면, H 는 G의 normal subgroup 이다.

(관찰 12.3.11.) \phi : G \to G’ 이 group homomorphism 이면, ker \phi \trianglelefteq G (그러나, 반드시 im \phi \trianglelefteq G’일 필요는 없다.

(정의 12.3.12.) N \trianglelefteq G 일 때, \pi : G \to G/N 을 \pi(x) = \bar{x} (x \in G) 라고 정의하고, \pi 를 natural projection 이라고 부른다.

(관찰 12.3.14.) Cyclic group 의 quotient group 은 cyclic 이다.

(표기법 12.3.16.) 0 < n \in \mathbb{Z} 일 때, \mathbb{Z}_{n} = \mathbb{Z}/n\mathbb{Z} 로 표기한다. 이 때, 연산은 \bar{a} + \bar{b} = \bar{a+b} 로 표기한다. (단, a,b \in \mathbb{Z}).

(관찰 12.3.17.) \mathbb{Z}{n} 은 cyclic group 이다. 따라서 \mathbb{Z}{n} \approx \mu_{n} 이다.

(정의 12.3.20.) G가 group 일 때, Z(G) = { z \in G | zg = gz for all g \in G} 로 표기하고, Z(G) 를 G의 center 라고 부른다.

(정의 12.3.24.) Projective general linear group PGL_{n}(F) 와 projective special linear group PSL_{n}(F) 를 각각 PGL_{n}(F) = GL_{n}(F) / Z(GL_{n}(F)) , PSL_{n}(F) = SL_{n}(F) / Z(SL_{n}(F)) 로 정의한다. PSO, PSU 등도 유사하게 정의할 수 있다.

(subsection 12.4.) Quotient space

(명제 12.4.1.) W \le V 일 때, coset space V/W 의 연산을 \bar{u} + \bar{v} = \bar{u+v} , c\bar{v} = \bar{cv} (u,v \in V, c \in F) 로 정의하면, V /W 는 vector space 가 된다. 우리는 V/W 를 [quotient space of V modulo W] 라고 부른다.

(정리 12.4.6.) V 가 f.d.v.s.이면 dim(V/W) = dim V = dim W 이다.

(subsection 12.5.) Isomorphism Theorem

(관차 12.5.1.) \phi : G \to H 가 group homomorphism 일 때, \phi (g) = h \in im \phi 이면, \phi^{-1}(h) = g \cdot ker \phi = \bar{g} 이다. (여기에서, (g \cdot ker \phi) 는 [G/ker \phi 의 coset] 을 뜻한다.)

(Group 의 First Isomorphism Theorem) (정리 12.5.2.) \phi : G \to H 가 group homomorphism 일 때, 함수 \bar{\phi} : G / ker \phi \to \im \phi 를 \bar{\phi} (\bar{g}) = \phi(g) , (g \in G) 로 정의하면, \bar{\phi} 는 well-defined 된 group isomorphism 이다. 즉, \begin{tikzpicture}\matrix (m) [matrix of math nodes,row sep=3em,column sep=4em,minimum width=2em] {G/ker \phi &im \phi \}; \path[-stealth] (m-1-1) edge node [above] {$\bar{\phi}$} node [below]{$\approx$} (m-1-2);\end{tikzpicture} 이다.

(벡터공간의 First Isomorphism Theorem) (정리 12.5.11.) L : V \to W 가 linear map 일 때, 함수 \bar{L} : V/ker L \to im L 을 \bar{L} (\bar{v}) = L(v) (v \in V) 로 정의하면, \bar{L} 은 well-defined 된 vector space isomorphism 이다. 즉, \begin{tikzpicture}\matrix (m) [matrix of math nodes,row sep=3em,column sep=4em,minimum width=2em] {V/ker L &im L \}; \path[-stealth] (m-1-1) edge node [above] {$\bar{\phi}$} node [below]{$\approx$} (m-1-2);\end{tikzpicture} 이다.

(관찰 12.5.14.) H \trianglelefteq G 이고 K \le G 이면, HK \le G 이다.

(Group 의 Second Isomorphism Theorem) H \trianglelefteq G 이고 G \le G 일 때, 함수 \bar{\jmath} : \frac{K}{H \cap K } \to \frac{HK}{H} 를 \bar{\jmath} (\bar{k}) = \bar{k} , (k \in K) 로 정의하면, \bar{\jmath} 는 well-defined 된 group isomorphism 이다. 따라서, \begin{tikzpicture}\matrix (m) [matrix of math nodes,row sep=3em,column sep=4em,minimum width=2em] {\frac{K}{H\capK} &\frac{HK}{H} \}; \path[-stealth] (m-1-1) edge node [above] {$\bar{\jmath}$} node [below]{$\approx$} (m-1-2);\end{tikzpicture} 이다.

(벡터공간의 Second Isomorphism Theorem) U, W \le V 일 때, 함수 \bar{\jmath} : \frac{W}{U \cap W } \to \frac{U + W}{W} 를 \bar{\jmath} (\bar{w}) = \bar{w} , (w \in W) 로 정의하면, \bar{\jmath} 는 well-defined 된 group isomorphism 이다. 따라서, \begin{tikzpicture}\matrix (m) [matrix of math nodes,row sep=3em,column sep=4em,minimum width=2em] {\frac{W}{U\capW} &\frac{U+W}{U} \}; \path[-stealth] (m-1-1) edge node [above] {$\bar{\jmath}$} node [below]{$\approx$} (m-1-2);\end{tikzpicture} 이다.

(따름정리 12.5.18.) V 가 f.d.v.s. 일 때, U,W \le V 이면, dim(U+W) = dim U + dim W – dim(U \cap W) 이다.

Triangularization

(Section 13) Triangularization

(subsection 13.1.) Triangularization

(정의) 이 장에서는 모든 것이 유한차원이라고 가정한다.

(관찰 13.1.1.) W \le V , W’ \le V’ 이고, linear map L : V \to V’ 이 LW \le W’ 을 만족한다고 가정할 때, \bar{L} : V/W \to V’/W’ 을 \bar{L}\bar{v} = \bar{Lv} (v \in V) 로 정의하면, \bar{L} 은 well-defined 된 linear map 이다. (이때, 우리는 L이 \bar{L} 을 naturally induce 한다고 말한다.)

(보조정리 13.1.4.) T \in \mathfrak{LM} 이고, W는 V의 T-invariant subspace 라고 하자. \bar{T} : \bar{V} \to \bar{V} 는 (관찰 13.1.1) 과 같다고 하고, \mathfrak{C} = {v_{1}, …, v_{m}} 은 W의 basis, \mathfrak{B} = {v_{1}, …, v_{n}} 은 V의 basis, 그리고 \bar{\mathfrak{B}} = {\bar{v_{m+1}}, …, \bar{v_{n}}} 는 \bar{V} 의 basis 라고 하자. 그러면, [T]{\mathfrak{B}}^{\mathfrak{B}} = $\left( \begin{array}{c|c} [T|{W}]{\mathfrak{C}}^{\mathfrak{C}} & * \ \hline 0 & [\bar{T}]{\bar{\mathfrak{B}}}^{\bar{\mathfrak{B}}}\end{array} \right)$

(Triangularization) (정리 13.1.6.) F = \mathbb{C} 이고 T \in \mathfrak{LM} 이면, [T]_{\mathfrak{B}}^{\mathfrak{B}} 가 upper-triangle matrix 인 V의 기저 \mathfrak{B} 가 존재한다.

(subsection 13.2.) Triangularization 의 결과

(따름정리 13.2.1.) T \in \mathfrak{LM} 일 때, T의 characteristic polynomial \phi_{T}(t) = t^{n} + a_{n-1} t^{n-1} + \cdots + a_{1} t + a_{0} (단, a_{1}, …, a_{n-1} \in F) 가 \mathbb{C}-위에서 \phi_{T}(t) = (t-\lambda_{1})(t-\lambda_{2}) \cdots (t-\lambda_{n}), (단, \lambda_{1}, …, \lambda_{n} \in \mathbb{C}) 로 인수분해되면, det(T) = \lambda_{1}\lambda_{2} \cdots \lambda_{n} = (-1)^{n}a_{0} , tr(T) = \lambda_{1} + \lambda_{2} + \cdots + \lambda_{n} = -a_{n-1} 이다.

(따름정리 13.2.2.) N \in \mathfrak{LM} 이 nilpotent 이면, N의 characteristic polynomial 은 \phi_{N}(t) = t^{n} 이다.

(정의 13.2.5.) T \in \mathfrak{LM} 일 때, (T-I) 가 nilpotent 이면 T를 unipotent operator (matrix) 라고 부른다.

(정의 13.2.9.) T,S \in \mathfrak{LM} 일 때, V 의 하나의 기저 \mathfrak{B} 에 관해 [T]{\mathfrak{B}}^{\mathfrak{B}} 와 [S]{\mathfrak{B}}^{\mathfrak{B}} 가 모두 upper-triangular matrix 이면, 우리는 T와 S가 simultaneously triangularizable 하다고 말한다. 여러 개 (무한개 포함) 의 경우도 마찬가지로 정의한다.

(명제 13.2.10.) T,S \in \mathfrak{LM} 이고 TS = ST 이면 – 즉 T 와 S 가 commute 하면 – T와 S는 common eigen-vector 를 갖는다. 즉, [Tv = \lambda v, Sv = \mu v for some \lambda , \mu \in \mathbb{C}] 인 non-zero vector v \in V 가 존재한다.

(명제 13.2.11.) T,S \in \mathfrak{LM} 이고, TS = ST 이면, T,S 는 simultaneously triangularizable 하다.

Bilinear Form

(Section 14) Bilinear form

(subsection 14.1) Bilinear form

(정리 14.1.6.) \mathfrak{B} 를 n-dimensional F-vector space V 의 고정된 기저라고 할 때, 함수 \Omega_{\mathfrak{B}} : \mathfrak{Bil}(V) \to \mathfrak{M}{n,n}(F) 를 \Omega{\mathfrak{B}}(B) = [B]{\mathfrak{B}} , (B \in \mathfrak{Bil}(V)) 로 정의하면 \Omega{\mathfrak{B}} 는 vector space isomorphism 이다. 이때, [B]_{\mathfrak{B}} 를 (기저 \mathfrak{B} 에 관한) B의 행렬이라고 부른다.

(관찰 14.1.8.) F^{n} 에 bilinear form B_{\mathcal{E}}^{J} 가 주어져 있다고 하자. J 가 가역일 때, A \in \mathfrak{M}{n,n}(F) 이면 B{\mathcal{E}}^{J}(AX,Y) = B_{\mathcal{E}}^{J}(X,(J^{-1} \cdot ^{t}A \cdot J) Y ), (X,Y \in F^{n})

(정의 14.1.9.) B : V \times V \to F 가 F-vector space V 의 bilinear form 일 때,

B가 다음 조건 B(v,w) = B(w,v) (v,w \in V) 를 만족하면, B를 symmetric bilinear form 이라고 부른다.

B가 다음 조건 B(v,v) = 0 (v \in V) 를 만족하면, B를 alternating bilinear form 이라고 부른다.

(관찰 14.1.10.) \mathfrak{B} 가 f.d.v.s. V의 기저이고, B 가 V의 bilinear form 일 때, B가 symmetric 이기 위한 필요충분조건은 [B]_{\mathfrak{B}} 가 symmetric matrix 인 것이다.

(subsection 14.2.)

(정의 14.2.2.) 유한차원 벡터공간 V위에 정의된 함수 Q : V \to F 가 다음 조건을 만족하면, Q를 V의 quadratic form 이라고 부른다.

B_{Q} : V \times V \to F 를 B_{Q} (v,w) = Q(v + w) -Q(v) – Q(w) (v,w \in V) 로 정의하면, B_{Q} 는 V의 symmetric bilinear form.

v \in V, c \in F 이면, Q(cv) = c^{2}Q(v)

따라서, quadratic form 이 주어지면, symmetric bilinear form 이 주어진다. 그리고, 그 역도 성립한다.

(관찰 14.2.4.) V가 유한차원 벡터공간을 때, 다음 대응관계 Q \leftrightarrow B_{Q} 는 [V의 quadratic form 전체의 집합] 과 [V의 symmetric bilinear form 전체 집합] 사이의 bijection 이다.

(정의 14.2.5.) 유한차원 벡터공간 V에 symmetric bilinear form B 가 주어졌다고 하자. 그러면, 앞 (관찰 14.2.4.) 에 의해, V의 quadratic form Q도 자동적으로 주어진다. 이 때, (V,B) = (V,B,Q) 를 quadratic space 라고 한다.

(section 14.3.) Orthogonal group 과 Symplectic group

(정의 14.3.1.) Quadratic space (V,B) 가 주어졌을 때 (따라서, B는 symmetric bilinear form 이고, quadratic form Q는 자동으로 주어져있다), O(V,B) = {L \in GL(V) | B(Lv, Lw) = B(v,w) for all v, w \in V} 로 정의하고, O(V,B) 를 (V,B) 로부터 만들어진 orthogonal group 이라고 부른다. 또, O(V,B) 의 원소는 물론 orthogonal operator 라고 부른다. 그리고, (V,B) 와 O(V,B) 를 공부하는 것을 orthogonal geometry 라고 부른다.

(명제 14.3.5.) \mathfrak{B} 를 quadratic space (V,B) 의 basis 라고 할 때, J = [B]_{\mathfrak{B}} 로 표기하면 (따라서 , J는 symmetric)

{[L]{\mathfrak{B}}^{\mathfrak{B}} \in GL{n}(F) | L \in O(V,B)} = {A \in GL_{n}(F) | ^{t}A \cdot J \cdot A = J} 이다.

(정의 14.3.6.) J \in \mathfrak{M}{n,n}(F) 가 symmetric matrix 일 때, O{n}^{J}(F) = {A \in GL_{n}(F) } ^{t}A \cdot J \cdot A = J} 로 표기하고, J에 대응하는 orthogonal group 이라고 부른다.

(정의 14.3.8.) B가 유한차원 벡터공간 V의 alternating bilinear form일 때, Sp(V,B) = {L \in GL(V) | B(Lv, Lw) = B(v,w) for all v , w \in V} 로 정의하고, Sp(V,B) 를 (V,B) 로부터 만들어진 symplectic group 이라고 부른다. 그리고 (V,B) 와 Sp(V,B) 를 공부하는 것을 symplectic grometry 라고 부른다.

(명제 14.3.9.) B가 유한차원 벡터공간 V의 alternating bilinear form 이고 , \mathfrak{B} 가 V의 기저일 때, J = [B]_{\mathfrak{B}} 로 표기하면 (따라서 ,J 는 skew-symmetric)

{[L]{\mathfrak{B}}^{\mathfrak{B}} \in GL{n}(F) | L \in Sp(V,B) } = {A \in GL_{n}(F) | ^{t}A \cdot J \cdot A = J} 이다.

(정의 14.3.10.) J \in \mathfrak{M}{n,n}(F) 가 skew-symmetric matrix 일 때, SP{n}^{J}(F) = {A \in GL_{n}(F) | ^{t}A \cdot J \cdot A = J} 로 표기하고, J에 대응하는 symplectic group 이라고 부른다.

(subsection 14.4.) O(1,1) 과 O(3,1)

(표기법) F = \mathbb{R}, J = diag(1,-1) 이라고 하면 \mathbb{R}^{2} 의 symmetric bilinear form B_{\mathcal{E}}^{J} 는 B_{\mathcal{E}}^{J} (\binom{a}{b} , \binom{c}{d} ) = ac – bd , (a,b,c,d \in \mathbb{R}) 로 주어질 것이다. 이 때, B_{\mathcal{E}}^{J} 에 대응하는 orthogonal group 을 O(1,1) = O_{2}^{J} (\mathbb{R}) = {A \in GL_{2}(\mathbb{R}) |^{t} A \cdot J \cdot A = J} 로 표기하자.

(정의 14.4.7.) J = diag(1,1,1,-1) 일 때, O(3,1) = O_{4}^{J}(\mathbb{R}) = {A \in GL_{4}(\mathbb{R}) | ^{t}A \cdot J \cdot A = J} 로 표기하고, O(3,1) 을 Lorenz group 이라고 부른다.

(subsection 14.5.) Non-degenerate Symmetric Bilinear form

(정의 14.5.1.) v, w \in V 일 때, B(v,w) = 0 이면 v \perp w 로 표기하고, v와 w 는 서로 수직이라고 말한다. 또, S,T \subseteq V 일 때, [B(v,w) = 0 for all v \in S, w \in T] 이면, S \perp T 라고 표기하고 S와 T는 서로 수직이라고 말한다.

(정의 14.5.3.) V의 basis \mathfrak{B} = {v_{i}} 가 mutually perpendicular subset 이면, \mathfrak{B} 를 V의 orthogonal basis 라고 부른다. 또, V의 orthogonal basis \mathfrak{B} = {v_{i}} 가 unit vector 들로 이루어져 있으면 (즉 ,B(v_{i}, v_{i}) = 1 for all i 이면) , \mathfrak{B} 를 V의 orthonormal basis 라고 부른다.

(표기법 14.5.5.) S \subseteq V 일 때, S^{\perp} = {v \in V | v \perp w for all w \in S} 로 표기한다.

(정의 14.5.9.) V^{\perp} = 0일 때, 우리는 B를 non-degenerate symmetric bilinear form 이라고 부른다.

(명제 14.5.11.) \mathfrak{B} 가 V의 basis 일 때, 다음은 동치이다.

B는 non-degenerate.

det([B]_{\mathfrak{B}}) \neq 0.

(정리 14.5.12.) B가 non-degenerate 일 때, W \le V 이면 dim W^{\perp} = dim V – dim W 가 성립한다.

(관찰 14.5.13.) [B(v,v) = 0 for all v \in V] 이면, B = 0 이다.

(보조정리 14.5.14.) w \in V 이고 B(w,w) \neq 0 이면, V = \langle w \rangle \oplus \langle w \rangle^{\perp} 이 성립한다.

(정리 14.5.15.) 모든 quadratic space (V,B) 는 orthogonal basis 를 갖는다. (물론 V \neq 0). (B 가 non-degenerate 일 필요도 없다.)

(정의 14.5.16.) 만약 , w \in V 가 B(w,w) \neq 0 을 만족하면, 함수 S_{w} : V \to V 를 S_{w}(v) = v - \frac{2B(v,w)}{B(w,w)} w, (v \in V) 라 정의하고, S_{w} 를 w에 관한 reflection 또는 symmetry 라고 부른다.

(subsection 14.6.) Dual space 와 Dual map

(명제 14.6.1.) 함수 \psi^{V} : V \to V^{} 를 (\psi^{V}(v))(f) = f(v) , (v \in V , f \in V^{*}) 로 정의하면, \psi^{V} 는 vector space isomorphism 이다. 이 때, \psi^{V} 는 [기저의 선택과 무관한 natural isomorphism] 이므로, 우리는 \psi^{V} 를 사용해 V와 V^{} 를 identify 한다.

(관찰 14.6.2.) v \in V 일 때, [f(v) = 0 for all f \in V^{*}] 이면, v = 0 이다.

(관찰 14.6.3.) {v_{1}, …, v_{n}} 이 f.d.v.s. V의 기저이면, \psi^{V}(v_{i}) = v_{i}^{} (i = 1,…,n) 이 성립한다. (단, v_{i}^{} = (v_{i}^{})^{} ).

(정의 14.6.5.) V,W 가 vector space 이고 L : V \to W 가 linear map 일 때, linear map L^{} : W^{} \to V^{} 를 L^{}(f) = f \bullet L , (f \in W^{}) 으로 정의하고, L^{} 를 L의 dual map 이라고 부른다.

(관찰 14.6.7.) \mathfrak{B} = {v_{1}, …, v_{n}} 을 V의 기저라고 하고, \mathfrak{C} = {w_{1}, …, w_{m}} 은 W의 기저라고 하자. 이 때, L \in \mathfrak{L}(V,W) 이면 [L^{}]_{\mathfrak{B}^{}}^{\mathfrak{C}^{*}} = ^{t}([L]_{\mathfrak{C}}^{\mathfrak{B}}) 가 성립한다.

(subsection 14.7.) Duality

(표기법) F-벡터공간 V 와 그 dual V^{} 에 대해 bilinear form \epsilon : V \times V^{} \to F : \epsilon(v,f) = f(v) , (v \in V, f \in V^{}) 로 정의한다. S \subseteq V , T \subseteq V^{} 일 때, S^{perp} = {f \in V^{*} | \epsilon(v,f) = 0 for all v \in S} , T^{perp} = {v \in V | \epsilon(v,f) = 0 for all f \in T} , 로 표기한다.

(정리 14.7.3.) V가 유한차원이고, W \le V, Y \le V^{*} 이면 다음이 성립한다.

dim W^{perp} = dim V – dim W.

dim Y^{perp} = dim V – dim Y.

(따름정리 14.7.4.) V가 유한차원이고, Y \le V^{} 라고 하자. 만약, Y \neq V^{} 이면, [f(v) = 0 for all f \in Y] 인 0 \neq v \in W 가 존재한다.

(subsection 14.8.) B-identification

(정리 14.8.1.) 함수 \phi^{V} = \phi_{B}^{V} : V \to V^{*} 를 (\phi^{V}(v))(w) = B(w,v) , (v,w \in V) 로 정의하면, \phi^{V} 는 linear map 이다. 이 때, 다음 조건은 동치이다.

B는 non-degenerate symmetric bilinear form.

\phi^{V} 는 monomorphism.

\phi^{V} 는 isomorphism.

V^{*} 의 모든 원소는 \phi^{V} (v) 의 꼴. (단, v \in V).

(관찰 14.8.3.) \mathfrak{B} 가 V의 기저이고, \mathfrak{B}^{} 를 \mathfrak{B} 의 dual basis 라고 하면, [\phi_{B}^{V}]_{\mathfrak{B}^{}}^{\mathfrak{B}} = [B]_{\mathfrak{B}}.

(표기법 14.8.4.) (V,B) 가 유한차원 non-degenerate quadratic space 일 때, (정리 14.8.1.) 의 isomorphism \phi^{V} 는 기저의 선택과 무관한 natural isomorphism 이다. 따라서 이번에도 \phi^{V} 를 통하여 V 와 V^{} 를 identify 하고, V = V^{} 로 표기법을 남용한다. 이를 B-identification 이라고 한다. 그리고, 혼동의 가능성이 없다면 \phi_{B}^{V} (v) = \phi_{v} (v \in V) 로 간략히 표기하기로 한다.

(명제 14.8.9.) (V,B) 가 유한차원 non-degenerate quadratic space 일 때, 함수 I \times \phi_{B}^{V} : V \times V \to V \times V^{*} 를 (I \times \phi_{B}^{V})(v,w) = (v, \phi_{w}) (v,w \in V) 로 정의하면, 다음 사각형 diagram

\begin{tikzpicture} \matrix (m) [matrix of math nodes,row sep=3em,column sep=4em,minimum width=2em] { V \times V & F\ V \times V^{*} & F \}; \path[-stealth] (m-1-1) edge node [left] {$I \times \phi_{B}^{V} $} node [right] {$=$} (m-2-1) edge node [above] {$B$} (m-1-2) (m-1-2) edge node [right] {$ I $} node [left] {$=$} (m-2-2) (m-2-1) edge node [below] {$\epsilon$} (m-2-2);\end{tikzpicture}

은 commutative diagram 이다. 즉, B-identification 의 관점에서는 B = e 이다. (따라서, “\perp = perp “ 라고 말할 수 있다.)

(정리 14.8.10.) (V,B) 를 유한차원 non-degenerate quadratic space 라고 하고, W \le V 라고 하자. 그러면, B-identification 의 관점에서는 W^{\perp} = W^{perp} 이다. 특별히, dim W^{\perp} = dim V – dim W 이다.

(subsection 14.9.) Transpose Operator

(정의 14.9.1) L \in \mathfrak{L}(V,W) 라고 하자. 이때, 선형사상 L의 (B와 C에 관한) transpose map ^{t}L \in \mathfrak{L}(V,W) 는 – B-identification 과 C-identification 의 관점에서 - ^{t}L = L^{} 로 정의된다. (L^{} 는 L의 dual map) . 즉, ^{t}L : W \to V 는 다음 사각형 diagram

\begin{tikzpicture} \matrix (m) [matrix of math nodes,row sep=3em,column sep=4em,minimum width=2em] { W & V\ W^{} \times V^{} & F \}; \path[-stealth] (m-1-1) edge node [left] {$ \phi_{C}^{W} $} node [right] {$=$} (m-2-1) edge node [above] {$^{t}L$} (m-1-2) (m-1-2) edge node [right] {$ \phi_{B}^{V} $} node [left] {$=$} (m-2-2) (m-2-1) edge node [below] {$L^{*}$} (m-2-2);\end{tikzpicture}

을 commutative diagram 으로 만들어주는 유일한 linear map 이다. 이를 식으로 표현한다면, ^{t}L = (\phi_{B}^{V})^{-1} \bullet L^{*} \bullet \phi_{C}^{W} 가 된다.

L \in \mathfrak{L}(V,V) 일 때 T의 transpose operator ^{t}L 을 ^{t}L = (\phi_{B}^{V})^{-1} \bullet L^{*} \bullet \phi_{B}^{W} 로 정의한다.

(관찰 14.9.2.) L,M \in \mathfrak{L}(V,V) 일 때, B(Lv, w) = B(Mv,w) , (v, w \in V) 이면, L = M 이다. (뒷자리에서도 마찬가지다.)

(명제 14.9.3.) L \in \mathfrak{L}(V,W) 이면, ^{t}L 은 다음 조건 C(Lv, w) = B(v, ^{t}Lw) , (v \in V , w \in W) 를 만족하는 유일한 linear map 이다. (B,C 는 symmetric 이므로, 이 조건은 C(w,Lv) = B(^{t}Lw, v) , (v \in V, w \in W) 와 동치이다.)

(명제 14.9.4.) \mathfrak{B,C} 를 각각 V,W의 고정된 기저라고 하자.

L \in \mathfrak{L}(V,W) 일 때, J_{\mathfrak{B}} = [B]{\mathfrak{B}} , J{\mathfrak{C}} = [C]{\mathfrak{C}} 로 표기하면, [^{t}L]{\mathfrak{B}}^{\mathfrak{C}} = (J_{\mathfrak{B}})^{-1} \cdot ^{t} ([L]{\mathfrak{C}}^{\mathfrak{B}}) \cdot J{\mathfrak{C}} 이다.

특별히, L \in \mathfrak{L}(V,V) 일 때, J = [B]{\mathfrak{B}} 로 표기하면, [^{t}L]{\mathfrak{B}}^{\mathfrak{B }} = J^{-1}\cdot ^{t} ([L]_{\mathfrak{B}}^{\mathfrak{B}}) \cdot J 이다.

(명제 14.9.6.) O(V,B) = {L \in GL(V) | ^{t}L \bullet L = I} .

(관찰 14.9.8.) t : \mathfrak{L}(V,W) \to \mathfrak{L}(W,V) 를 t(L) = ^{t}L (L \in \mathfrak{L}(V,V)) 로 정의하면, t 는 isomorphism 이다. 특별히, 다음이 성립한다.

L, M \in \mathfrak{L}(V,W) 이면, ^{t}(L+M) = ^{t}L + ^{t}M

L \in \mathfrak{L}(V,W), c \in F 이면, ^{t}(cL) = c \cdot ^{t}L.

(관찰 14.9.9.) 다음이 성립한다.

^{t}I = I.

L \in \mathfrak{L}(V,W) 이고 M \in \mathfrak{L}(W,U) 이면, ^{t}(M \bullet L) = ^{t}L \bullet ^{t}M.

L \in \mathfrak{L}(V,W) 이면, ^{t}(^{t}L) = L.

Hermitian form

(Section 15) Hermitian form

(subsection 15.1.) Hermitial form

(정의 15.1.1.) V를 \mathbb{C}-vector space 라고 하자. 이 때, 함수 H : V \times V \to \mathbb{C} 가 , 모든 v,u,w \in V , c \in \mathbb{C} 에 대해, 다음 조건

H(u+v, w) = H(u,w) + H(v,w)

H(cv, w) = cH(v,w)

H(v,w) = \bar{H(w,v)}

를 만족하면, H를 V의 Hermitian form 이라고 부르고, (V,H) 를 Hermitian space 라고 부른다.

(정의) V가 유한차원 \mathbb{C}-벡터공간일 때, \mathfrak{Ker}(V) = {H : V \times V \to \mathbb{C} | H 는 Hermitian form} 이라고 표기하자.

(정의) (i,j)-성분이 H(v_{i}, v_{j}) 인 (n \times n)-행렬 [H]{\mathfrak{B}} = (H(v{i},v_{j})) 를 (기저 \mathfrak{B} 에 관한) H의 행렬이라고 부른다

(정리 15.1.6.) 함수 \Omega_{\mathfrak{B}} : \mathfrak{Ker}(V) \to {J \in \mathfrak{M}{n,n} (\mathbb{C}) | J 는 self-adjoint} 를 \Omega{\mathfrak{B}}(H) = [H]{\mathfrak{B}} (H \in \mathfrak{Ker}(V)) 로 정의하면 \Omega{\mathfrak{B}} 는 bijection 이다.

(명제 15.1.11.) J = [H]{\mathfrak{B}} 로 표기하면(따라서, J 는 self-adjoint) {[L]{\mathfrak{B}}^{\mathfrak{B}} \in GL_{n}(\mathbb{C}) | L \in U(V,H) } = {A \in GL_{n}(\mathbb{C}) | ^{t}A \cdot J \cdot \bar{A} = J} 이다.

(정의 15.1.12.) J \in \mathfrak{M}{n,n}(\mathbb{C}) 가 self-adjoint 일 때, U{n}^{J}(\mathbb{C}) = {A \in GL_{n}(\mathbb{C}) | ^{t}A \cdot J \cdot \bar{A} = J} 로 표기하고, J 에 대응하는 unitary group 이라고 부른다.

(subsection 15.2.) Non-degenerate Hermitian form

(정의 15.2.1.) v,w \in V 일 때, H(v,w) = 0 이면, v \perp w 로 표기하고 v 와 w는 서로 수직이라고 말한다. 그리고, S,T \subseteq V 일 때, [H(v,w) = 0 for all v \in S, w \in T] 이면, S \perp T 로 표기하고 S와 T는 서로 수직이라고 말한다. 또, S^{\perp} = {v \in V | v \perp w for all w \in S} 로 표기한다.

(정의 15.2.2.) [H]{\mathfrak{B}} 가 대각행렬이면, \mathfrak{B} 를 V의 orthogonal basis 라고 부른다. 또, [H]{\mathfrak{B}} = I 이면, \mathfrak{B} 를 V의 orthonormal basis 라고 부른다.

(정의 15.2.4.) V^{\perp} = 0 일 때, 우리는 H를 non-degenerate Hermitian form 이라고 부른다.

(명제 15.2.5.) H가 non-degenerate 이기 위한 필요충분조건은 det([H]_{\mathfrak{B}}) \neq 0 인 것이다.

(정리 15.2.6.) H가 non-degenerate 일 때, W \le V 이면 dim W^{\perp} = dim V – dim W 가 성립한다.

(관찰 15.2.7.) [H(v,v) = 0 for all v \in V] 이면, H = 0 이다.

(정리 15.2.9.) 모든 non-zero Hermitian space (V,H) 는 orthogonal basis 를 갖는다.

(subsection 15.3.) H-Identification 과 Adjoint operator

(정의) 함수 \phi^{V} = \phi_{H}^{V} : V \to V^{*} 를 (\phi^{V} (v)) (w) = H(w,v) (v,w \in V ) 로 정의하자.

(정리 15.3.3.) 함수 \phi^{V} = \phi_{H}^{V} : V \to V^{} 에 대해 H가 non-degenerate Hermitian form 이면, \phi^{V} 는 bijection이다. 특별히, V^{} 의 모든 원소는 \phi^{V}(v) 의 꼴로 쓸 수 있다. (단, v \in V)

(관찰 15.3.4.) W \le V 이면, dim W = dim \phi_{H}^{V}(W) 이다.

(표기법 15.3.5.) (정리 15.3.3.) 의 bijection \phi_{H}^{V} 는 기저의 선택과 무관한 natural bijection 이다. 따라서, B-identification 과 유사하게 \phi_{H}^{V} 를 통하여 V 와 V^{} 를 identify 하고, V = V^{} 로 표기법을 쓴다. 이를 우리는 H-identification 이라고 부른다. 그리고 혼동의 가능성이 없다면 \phi_{H}^{V}(v) = \phi_{v} (v \in V) 라고 쓰기도 한다.

(정리 15.3.9.) W \le V 라고 하자. 그러면, H-identification 의 관점에서는 W^{\perp} = W^{perp} 이다. 특별히, dim W^{\perp} = dim V – dim W 이다.

(정의 15.3.10.) L \in \mathfrak{L}(V,W) 이라고 하자. 이 때, L의 (H와 K에 관한) adjoint map L^{\star} \in \mathfrak{L}(W,V) 는 – H-identification 과 K-identification 의 관점에서 – L^{\star} = L^{*} 로 정의된다. 즉, L^{\star} : W \to V 는 다음 사각형 diagram

\begin{tikzpicture} \matrix (m) [matrix of math nodes,row sep=3em,column sep=4em,minimum width=2em] { W & V\ W^{} \times V^{} & F \}; \path[-stealth] (m-1-1) edge node [left] {$ \phi_{K}^{W} $} node [right] {$=$} (m-2-1) edge node [above] {$L^{\star}$} (m-1-2) (m-1-2) edge node [right] {$ \phi_{H}^{V} $} node [left] {$=$} (m-2-2) (m-2-1) edge node [below] {$L^{*}$} (m-2-2);\end{tikzpicture}

을 commutative diagram 으로 만들어주는 유일한 linear map이다. 이를 식으로 표현한다면, L^{} = (\phi_{H}^{V})^{-1} \bullet L^{} \bullet \phi_{K}^{W} 가 된다.

(명제 15.3.12.) L \in \mathfrak{L}(V,W) 이면, L^{} 는 다음 조건 K(Lv, w) = H(v, L^{}w), (v \in V , w \in W) 를 만족하는 유일한 linear map 이다. (이 조건은 K(w,Lv) = H(L^{*}w, v) (v \in V, w \in W) 와 동치이다.)

(명제 15.3.13.) L \in \mathfrak{L}(V,W) 일 때, J_{\mathfrak{B}} = [H]{\mathfrak{B}} , J{\mathfrak{C}} = [K]{\mathfrak{C}} 로 표기하면, [L^{*}]{\mathfrak{B}}^{\mathfrak{C}} = \bar{J_{\mathfrak{B}}}^{-1} \cdot ([L]{\mathfrak{B}}^{\mathfrak{C}})^{*} \cdot \bar{J{\mathfrak{C}} 이다.

특별히, L \in \mathfrak{L}(V,V) 일 때, J = [B]{\mathfrak{B}}로 표기하면, [L^{*}]{\mathfrak{B}}^{\mathfrak{B}} = \bar{J}^{-1} \cdot ([L]_{\mathfrak{B}}^{\mathfrak{B}})^{*} \cdot \bar{J}

(명제 15.3.14.) U(V,H) = { L \in GL(V) | L^{*} \bullet L = I}

(관찰 15.3.16.) L,M \in \mathfrak{L}(V,W) 이고 c \in \mathbb{C} 이면, 다음이 성립한다.

(L + M)^{} = L^{} + M^{*}.

(cL)^{} = \bar{c} \cdot L^{}.

(subsection 15.4.) 왜 Non-degenerate 인 경우만?

(관찰 15.4.1.) L \in U(V,H) 이면, V^{\perp} 는 V의 L-invariant subspace 이다. 다시 말해, L(V^{\perp}) = V^{\perp} 이고, 따라서 L|_{V^{\perp}} \in GL(V^{\perp}) 이다.

(관찰 15.4.2.) \bar{H} 는 well-defined 된 \bar{V} 의 Hermitian form 이다.

(관찰 15.4.3.) \bar{H} 는 non-degenerate 이고, [\bar{H}]{\bar{\mathfrak{B}} = [H|{W \times W} ]_{\mathfrak{D}} 이다.

(명제 15.4.5.) 함수 \Lambda : U(V,H) \to GL(V^{\perp}) \times \mathfrak{M}{m,n-m}(\mathbb{C}) \times U(\bar{V}, \bar{H}) 를 \Lambda(L) = (L|{V^{\perp}} , [L]_{\mathfrak{C}}^{\mathfrak{D}} , \bar{L}) , (L \in U(V,H)) 로 정의하면, \Lambda 는 bijection 이다.

Spectral Theorem

(Section 16) Spectral Theorem

(subsection 16.1) 표기법과 용어

(표기법 16.1.1.) 이 책에서 [T \in \mathfrak{LM}] 이라고 하면, 항상 [T \in \mathfrak{L}(V,V) 혹은 T \in \mathfrak{M}{n,n}(F)] 를 뜻하기로 약속한다. 이때, dim V = n 이라고 놓자. 물론 T가 행렬이면 V = F^{n} 으로 이해하고, T = L{T} 로 혼동한다. (즉, 행렬은 특수한 선형사상으로 이해한다.) 또, V \neq 0 이라고 가정한다.

(표기법 16.1.2.) F = \mathbb{C} 이고 (V,H) 가 non-degenerate Hermitian space 일 때, T = L \in \mathfrak{L}(V,V) 이면, T^{} = L^{} 로 이해한다. 따라서, T = A \in \mathfrak{M}{n,n}(\mathbb{C}) 이면 T^{*} = (L{A})^{*} 로 이해한다는 뜻이 된다.

(F에 제한은 없지만 대부분의 경우에 F = \mathbb{R} 이고, ) (V,B) 가 non-degenerate quadratic space 일 때에는, T = L \in \mathfrak{L}(V,V) 이면, T^{} = ^{t}L 로 이해한다. 따라서, T = A \in \mathfrak{M}_{n,n}(F) 이면 T^{} = ^{t}(L_{A}) 로 이해한다는 뜻이 된다. 그리고 quadratic space 임을 강조하기 위하여, T^{*} 보다는 ^{t}T 의 표기법을 더 자주 사용한다.

(관찰 16.1.4.) T \in \mathfrak{LM} 일 때, V의 subspace W 가 T-invariant 이고 동시에 T^{}-invariant 이면, (T|_{W})^{} = T^{*}|_{W} 이다.

(정의 16.1.5.) T \in \mathfrak{LM} 일 때

만약 [T^{*} = T] 이면, (V,H) 의 경우에는 T를 self-adjoint operator(또는, Hermitian operator) 라고 부르고, (V,B)의 경우에는 T를 symmetric operator 라고 부른다.

[T^{*} \cdot T = I] 이면 ,(V,H)의 경우에는 T를 unitary operator 라고 부르고, (V,B) 의 경우에는 T를 orthogonal operator 라고 부른다.

[T \cdot T^{} = T^{} \cdot T] 이면, T를 normal operator 라고 부른다.

(subsection 16.2.) Normal operator

(표기법 16.2.1.) 이 절에서는 항상 F = \mathbb{C} 이고, (V,\langle , \rangle ) 는 Hermitian inner product \langle, \rangle 가 주어진 n-차원 \mathbb{C}-vector space 이다.

(관찰 16.2.2.) T \in \mathfrak{LM} 이라고 할 때, [T의 eigen-vector 들로 이루어진 V의 orthonormal basis] 가 존재하려면, T는 normal 이어야만 한다.

(관찰 16.2.3.) T \in \mathfrak{LM} 일 때, 다음은 동치이다.

T 는 normal.

\langle Tv, Tw \rangle = \langle T^{}v, T^{}w \rangle for all v, w \in V.

(관찰 16.2.4.) T \in \mathfrak{LM} 이 normal이고, \lambda \in \mathbb{C} 이면, (T-\lambda I) 도 normal 이다.

(관찰 16.2.5) T \in \mathfrak{LM} 이 normal 일 때, Tv = \lambda v 이면 (단, 0 \neq v \in V, \lambda \in \mathbb{C}) , 즉 v가 eigen-value \lambda 를 갖는 T의 eigen-vector 이면, T^{}v = \bar{\lambda} v 이다. 즉, v는 eigen-value \bar{\lambda} 를 갖는 T^{} 의 eigen-vector 이다.

(Spectral Theorem ; positive definite Hermitian case) (정리 16.2.6.) T \in \mathfrak{LM} 일 때, 다음 조건은 동치이다.

T 는 normal.

[T의 eigen-vector 들로 이루어진 V의 orthonormal basis] 가 존재.

(정의 16.2.7.) The group of diagonal unitary matrices T_{U(n)} 을 T_{U(n)} = U(n) \cap D_{n}(\mathbb{C}) 로 정의한다. 그리고, T_{SU(n)} = SU(n) \cap D_{n}(\mathbb{C}) = T_{U(n)} \cap SL_{n}(\mathbb{C}) 로 정의한다.

(Spectral Theorem for U(n)) (따름정리 16.2.9.) 만약 A \in U(n) 이라고 하면, U^{-1}AU \in T_{U(n)} 인 U \in U(n) 이 존재한다. 따라서, U(n) = \bigcup_{U \in U(n)} U \cdot T_{U(n)} \cdot U^{-1} 로 쓸 수 있다.

(Spectral Theorem for SU(n)) (따름정리 16.2.10.) 만약 A \in SU(n) 이라고 하면, U^{-1}AU \in T_{SU(n)} 인 U \in SU(n) 이 존재한다. 따라서, SU(n) = \bigcup_{U \in SU(n)} U \cdot T_{SU(n)} \cdot U^{-1} 로 쓸 수 있다.

(subsection 16.3.) Symmetric Operator

(표기법 16.3.1.)이 절에서는 항상 F = \mathbb{R} 이고, (V,\langle , \rangle ) 는 inner product \langle, \rangle 가 주어진 n-차원 \mathbb{R}-vector space 이다. 그리고 T^{*} 보다 ^{t}T 를 더 많이 사용한다.

(명제 16.3.2.) T \in \mathfrak{LM} 이 symmetric operator 이면, T는 항상 (real) eigen-value 를 갖는다.

Symmetric matrix A \in \mathfrak{M}_{n,n}(\mathbb{R}) 은 항상 (real) eigen-value 를 갖는다.

(관찰 16.3.4.)T \in \mathfrak{LM} 이라고 할 때, [T의 eigen-vector 들로 이루어진 V의 orthonormal basis] 가 존재하려면 , T는 symmetric 이어야만 한다.

(Spectral Theorem ; positive definite real quadratic case) (정리 16.3.5.) T \in \mathfrak{LM} 일 때, 다음 조건은 동치이다.

T는 symmetric.

[T의 eigen-vector 들로 이루어진 V의 orthonormal basis] 가 존재.

(Spectral Therorem for real symmetric matrices) (따름정리 16.3.6.) \mathbb{R}^{n} 에 dot product 가 주어졌을 때, A \in \mathfrak{M}_{n,n}(\mathbb{R}) 이면, 다음 조건은 동치ㅣ다.

A는 symmetric matrix.

[A의 eigen-vector 들로 이루어진 \mathbb{R}^{n}의 orthonormal basis] 가 존재.

[U^{-1}AU 가 diagonal matrix] 인 U \in O(n) 이 존재.

(subsection 16.4.) Orthogonal Operator

(명제 16.4.1.) F = \mathbb{R} 이고, T \in \mathfrak{LM} 이면, [dim W = 1 혹은 dim W = 2] 인 V의 T-invariant subspace W 가 존재한다.

(Spectral Theorem for real orthogonal operators) (정리 16.4.2.) V를 inner product \langle , \rangle 가 주어진 finite dimensional \mathbb{R} – vector space 라 할때, T \in \mathfrak{LM} 이 orthogonal operator 이면, 다음 조건

V = W_{1} \oplus \cdots \oplus W_{s}.

W_{1}, …, W_{s} 는 V의 T-invariant subspace.

W_{1}, …., W_{s} 는 mutually orthogonal. 즉, W_{i} \perp W_{j} if i \neq j.

dim W_{i} = 1 or 2 for all i = 1,…,s

를 만족하는 V의 subspace W_{1}, …, W_{s} 가 존재한다.

(표기법) B_{i} 가 square matrix들일 때, diag(B_{1}, …, B_{r}) 은 i-th diagonal block 에 B_{i} 가 있는 행렬, 즉 diag(B_{1}, … , B_{r}) = \begin{pmatrix} B_{1} & 0 & \cdots & 0 \ 0 & B_{2} & \cdots & 0 \ \vdots & \vdots & \ddots & \vdots \ 0 & 0 & \cdots & B_{r} \end{pmatrix} 을 의미하기로 한다.

(정의 16.4.3.) O(n) 의 subgroup T_{\mp} 를

n이 홀수이면 T_{\mp} = {diag(\mp 1, A_{1}, …, A_{r}) \in O(n) | A_{1}, …, A_{r} \in SO(2)} , 로 정의하고

n이 짝수이면 T_{\mp} = {diag(A_{0}, A_{1}, …, A_{r}) \in O(n) | A_{0} \in O(2) and A_{1}, …, A_{r} \in SO(2)} 로 정의한다.

(정의 16.4.4.) SO(n) 의 subgroup T_{SO(n)} 을 T_{SO(n)} = SO(n) \cap T_{\mp} 로 정의한다.

(Spectral Theorem for O(n)) (따름정리 16.4.6.) 만약 A \in O(n) 이라고 하면, U^{-1}AU \in T_{\mp} 인 U \in O(n) 이 존재한다. 따라서, O(n) = \bigcup_{U \in O(n)} U \cdot T_{\mp} \cdot U^{-1} 로 쓸 수 있다.

(Spectral Theorem for SO(n)) (따름정리 16.4.8.) A \in SO(n) 이라고 하면, U^{-1}AU \in T_{SO(n)} 인 U \in SO(n) 이 존재한다. 따라서, SO(n) = \bigcup_{U \in SO(n)} U \cdot T_{SO(n)} \cdot U^{-1} 로 쓸 수 있다.

(subsection 16.5.) Non-Degenerate Case

새로운 발상, 새로운 언어, 새로운 도구가 필요하다.

(subsection 16.6) Epilogue

(Schur’s Theorem) (정리 16.6.1.) F = \mathbb{C} 이고 (V , \langle , \rangle) 는 n-차원 Hermitian inner product space 라고 할 때, T \in \mathfrak{LM{ 이면 [T]_{\mathfrak{B}}^{\mathfrak{B}} 가 upper-triangular matrix 인 V의 orthonormal basis \mathfrak{B} 가 존재한다.

(정의 16.6.2.) (표기법 12.3.16) 의 abelian group \mathbb{Z}{2} = {\bar{0}, \bar{1}} 에 다음과 같이 \bar{0} \cdot \bar{0} = \bar{0} \cdot \bar{1} = \bar{1} \cdot \bar{0} = \bar{0} , \bar{1} \cdot \bar{1} = \bar{1} 곱셈을 정의하면 \mathbb{Z}{2} 는 field 가 된다.

(정의 16.6.3.) n-dimensional \mathbb{Z}{2} -vector space (\mathbb{Z}{2}^{n}) 의 \mathbb{Z}{2}=subspace C 를 우리는 [linear (binary) code of length n] 이라고 부른다. 또, (\mathbb{Z}{2}^{n})에 ‘보통 내적’ B_{\mathcal{E}}^{I} 가 주어졌다고 할때, C^{\perp} 를 C의 dual code 라고 부른다.

(정리 16.6.4.) C 가 (\mathbb{Z}_{2})^{n} 의 linear code 이면, dim C + dim C^{\perp} = n 이다. 따라서, dim C = m 이라고 하면, |C| = 2^{m} 이고, |C^{\perp}| = 2^{n-m} 이다.

Topology Introduction

(Section 17) Topology 맛보기

(정의 17.1.8.) X \subseteq \mathbb{R}^{r} , Y \subseteq \mathbb{R}^{s} 일 때, 함수 f : X \to Y 가 다음 조건

f 는 bijection.

f 는 continuous 이고, 그의 역함수 f^{-1} 도 continuous.

를 만족하면 ,f 를 같은 생김새 함수라고 부르기로 한다. 이 때 X와 Y는 서로 같은 생김새라고 말한다.

(관찰 17.1.17)H 가 matrix group G \le GL_{n}(F) 의 subgroup 일 때, g \in G 이면, H 와 left coset g는 같은 생김새이다.

(정의 17.1.18) ([F,F’= \mathbb{R} or \mathbb{C}] 로 놓자. 즉, F \new F’ 일 수도 있다는 뜻.) G \le GL_{n}(F) \subset \mathbb{R}^{N} , H \le GL_{m}(F’) \subset \mathbb{R}^{M} 이 matrix group 일 때, 함수 \phi : G \to H 가 다음 조건

\phi 는 group isomorphism.

\phi 는 같은 생김새 함수.

를 만족하면, \phi 를 matrix group isomorphism 이라고 부르기로 한다. 이 때, G와 H는 서로 matrix group 으로서 isomorphic 하다고 말하고, \begin{tikzpicture}\matrix (m) [matrix of math nodes,row sep=3em,column sep=4em,minimum width=2em] {G &H \}; \path[-stealth] (m-1-1) edge node [above] {$\approx$} node [below]{$matrix gp$} (m-1-2);\end{tikzpicture} 으로 표기하자.

(관찰 17.1.21.) H가 matrix group G \le GL_{n}(F) 의 subgroup 일 때, g \in G 이면 , \begin{tikzpicture}\matrix (m) [matrix of math nodes,row sep=3em,column sep=4em,minimum width=2em] {H &gHg^{-1} \}; \path[-stealth] (m-1-1) edge node [above] {$\approx$} node [below]{$matrix gp$} (m-1-2);\end{tikzpicture} 이다.

(subsection 17.2.) Compactness 와 Connectedness

(명제 17.2.1.) X \subseteq \mathbb{R}^{N}, Y \subseteq \mathbb{R}^{M} 이고 f : X \to Y가 연속함수이면, 다음이 성립한다.

X가 compact 이면, f(X) 도 compact.

X 가 connected 이면, f(X) 도 connected.

(명제 17.2.9.) X_{i} \subseteq \mathbb{R}^{N_{i}} 일 때, X_{1} \times \cdot \times X_{r} 을 \mathbb{R}^{N_{1} + \cdot + N_{r}} 의 subset으로 생각하자.

모든 X_{i} 가 compact 이면 X_{1} \times \cdot \times X_{r} 도 compact.

모든 X_{i} 가 connected이면 X_{1} \times \cdot \times X_{r} 도 connected.

(정의 17.2.10) \mathbb{R}^{N} 의 subset T가 S^{1} \times \cdots \times S^{1} 과 같은 생김새이면, T를 torus 라고 부른다. Torus 는 compact connected group 이다.

(Spectral Theorem 의 따름정리) (따름정리 17.2.12.) U(n) 과 SU(n), 그리고 SO(n) 은 connected group 이다.

(관찰 17.2.13.) SO(n) 은 O(n) 의 connected component 이다.

O(n) 의 coset decomposition O(n) = SO(n) \mathfrak{U} (O(n) – SO(n)) 은 동시에 O(n) 의 [connected component decomposition] 이다.

(관찰 17.2.14.) SO^{\bullet}(1,1) 은 O(1,1) 의 connected component 이다.

[coset decomposition of O(1,1) modulo SO^{\bullet} (1,1)] 은 동시에 O(1,1) 의 [connected component decomposition] 이다.

Abstract Algebra

Simplex

(Def) Two sets, or topological spaces, are structurally same if there is a one-to-one function mapping one onto the other such that both this function and its inverse are continuous. Two spaces that are structurally same in this sense are homeomorphic.

(Def) oriented 0-simplex is a point P. Oriented 1-simplex is a directed line segment P_{1} P_{2} joining the points P_{1} and P_{2} and viewed as traveled in the direction from P_{1} to P_{2}. Oriented 2-simplex is a triangular region P_{1} P_{2} P_{3}.

P_{i} P_{j} P_{k} is equal to P_{1} P_{2} P_{3} if \begin{pmatrix} 1 & 2 & 3 \ i & j & k \end{pmatrix} is an even permutation. opposite to P_{1} P_{2} P_{3} if it is an odd permutation.

(Def) Oriented 3-simplex is given by an ordered sequence P_{1} P_{2} P_{3} P_{4} of four vertices of a solid tetrahedron. Simplexes are oriented, or have an orientation, meaning that we are concerned with the order of the vertices as well as actual points where vertices are located.

(Def) Boundary of a 0-sumplex P is an empty simplex. notation is “ \partial_{0} (P) = 0”

Boundary of a 1-simplex P_{1} P_{2} is defined by \partial_{1} (P_{1} P_{2}) = P_{2} – P_{1}.

Boundary of a 2-simplex is defined by \partial_{2} (P_{1} P_{2} P_{3}) = P_{2} P_{3} – P_{1} P_{3} + P_{1} P_{2}

Boundary of a 3-simplex is \partial_{3} (P_{1} P_{2} P_{3} P_{4}) = P_{2} P_{3} P_{4} – P_{1} P_{3} P_{4} + P_{1} P_{2} P_{4} – P_{1} P_{2} P_{3} . Similar definition holds for \partial_{n} for n > 3.

Each individual summand of the boundary of a simplex is a face of the simplex.

(Def) A space divided up into simplexes according to following requirements is a simplical complex.

Each point of the space belongs to at least one simplex.

Each point of the space belongs to only a finite number of simplexes.

Two different (up to orientation) simplexes either have no points in common or one is (except possibly for orientation) a face of the other or a face of a face of the other, etc. or the set of points in common is a face, or a face of a face, etc.,, of each complex.

(Def) For a simplical complex X, The group C_{n} (X) of (oriented) n-chains of X is the free abelian group generated by the (oriented) n-simplexes of X.

Every element of C_{n} (X) is a finite sum of the form \sum_{i} m_{i} \sigma_{i}, where the \sigma_{i} are n-simplexes of X and m_{i} \in \mathbf{Z}. We accomplish addition of chains by taking the algrbraic sum of the coefficients of each occurrence in the chains of a fixed simplex.

(Def) if \sigma is an n-simplex, \partial_{n} ( \sigma ) \in C_{n-1} (X) for n = 1,2,3. define C_{-1} (X) = {0} , then we will also have \partial_{0} (\sigma ) \in C_{-1} (X). Since C_{n} (X) is free abelian, \partial_{n} gives a unique boundary homomorphism \partial_{n} mapping C_{n} (X) into C_{n-1} (X) for n = 0,1,2,3.

(Def) Kernel of \partial_{n} consists of those n-chains with boundary 0. The elements of the kernel are n-cycles. The usual notation for the kernel of \partial_{n} , group of n-cycles, is ‘Z_{n} (X) ‘.

Image under \partial_{n} , the group of (n-1)-boundaries, consists of those (n-1)-chains that are boundaries of n-chains. This groups is denoted by ‘B_{n-1} (X) ’.

(Thm 41.9) Let X be a simplical complex, and let C_{n} (X) be the n-chains of X for n = 0,1,2,3. Then the composite homomorphism \partial_{n-1} \partial_{n} mapping C_{n} (X) into C_{n-2} (X) maps everything into 0 for n = 1,2,3. That is, for each c \in C_{n} (X) we have \partial_{n-1} (\partial_{n} (c)) = 0. We use the notation \partial_{n-1} \partial_{n} = 0, or \partial^{2} = 0.

(Cor 41.10) For n = 0,1,2 and 3, B_{n} (X) is a subgroup of Z_{n} (X).

(Def 41.11) The factor group H_{n} (X) = Z_{n} (X) / B_{n} (X) is the n-dimensional homology group of X.

Computations of Homology Groups

(Section 42) Computations of Homology Groups

(Def) If a space is divided up into pieces in such a way that near each point the space can be deformed to look like a part of some Euclidean space \mathbb{R}^{n} and the pieces into which the space was divided appear after this deformation as part of a simplical complex, then the original division of the space is a triangulation of the space.

homology groups of the space are defined formally as in the last section.

(Prop) invariance properties of homology groups.

The homology groups of a space are defined in terms of a triangulation, but is the same (isomorphic) groups no matter how the space is trangulated.

If one triangulated space is homeomorphic to another, the homology groups of the two spaces are the same (isomorphic) in each dimension n.

(Def) n-sphere S^{n} is the set of all points a distance of 1 unit from the origin in (n+1)-dimensional Euclidean space \mathbb{R}^{n+1}. The n-cell or n-ball E^{n} is the set of all points in \mathbb{R}^{n} a dinstance \le 1 from the origin.

(subsection) Connected and Contractable Spaces

(Def) A space is connected if any two points in it can be joined by a path lying totally in the space. If a space is not connected, it is split up into a number of pieces, each of which is connected buy no two of which can be joined by a path in space. These pieces are the connected components of the space.

(Thm 42.4) If a space X is trangulated into a finite number of simplexes, then H_{0} (X) is isomorphic to \mathbb{Z} \times \mathbb{Z} \times \cdots \times \mathbb{Z}, and the Betti number m of factors \mathbb{Z} is the number of connected componenets of X.

(Def) a space is contractible if it can be compressed to a point without being torn or cut, but always kept within the space it originally occupied.

(Thm 42.6) If X is a contractible space triangulated into a finite number of simplexes, then H_{n} (X) = 0 for n \ge 1.

E^{n} is contractable for n \ge 1, so H_{1} (E^{n}) = 0 for i > 0.

(prop) For n > 0, H_{n} (S^{n}) and H_{0}(S^{n}) are isomorphic to \mathbb{Z}, while H_{i} (S^{n}) = 0 for 0 < i < n.

(Def) an element of H_{n} (X), that is, a coset of B_{n} (X) in Z_{n} (X), a homology class. Cycles in the same homology class are homologous.

(Section 43) More Homology computations and applications

(Def) one-sided closed surface, the Klein bottle. the 1-dimensional homology group will have a nontrivial torsion subgroup reflecting the twist in the surface.

(subsection) Euler Characteristic

(Def) Let X be a finite simplical complex consisting of simplexes of dimension 3 and less, Let n_{0} be the total number of vertices in the triangulation, n_{1} the number of edges, n_{2} the number of 2-simplexes, and n_{3} the number of 3-simplexes. The number n_{0} – n_{1} + n_{2} – n_{3} = \sum_{i = 0}^{3} (-1)^{i} n_{i} is same no matter how the space X is triangulated. This number is the Euler characteristic \chi (X) of the space.

(Thm 43.7) Let X be a finite simplical complex (or triangulated space) of dimension \le 3. Let \chi (X) be the Euler characteristic of the space X, and let \beta_{j} be the Betti number of H_{j} (X). Then \chi (X) = \sum_{j = 0}^{3} (-1)^{j} \beta_{j} . This theorem holds also for X of dimension greater than 3, with the extension of the definition of the Euler characteristic to dimension greater than 3.

(subsection) Mapping of Spaces

(prop) a continuous function f mapping a space X into a space Y gives rise to a homomorphism f_{*n} mapping H_{n} (X) into H_{n} (Y) for n >0.

(prop) If z \in \mathbb{Z}{n} (X), and if f(z), regarded as the result of picking up z and setting it down in Y in the naively obvious way, should be an n-cycle in Y, then f{*n} (z + B_{n} (X)) = f(z) + B_{n} (Y). That is, if z represents a homology class in H_{n} (X) and f(z) is an n-cycle in Y, then f(z) represents the image homology class under f_{*n} of the homology class containing z.

(Def) a fixed point is some x \in E^{n} s.t. f(x) = x.

(Brower Fixed-point Theorem) (Thm 43.12) A continuous map f of E^{n} into itself has a fixed point for n \ge 1.

Homological Algebra

(Section 44) Homological Algebra

(subsection) Chain complexes and Mappings

(Def) For a simplical complex X, we get chain groups C_{k} (X) and maps \partian_{k} , as indicated in the diagram.

with \partial_{k-1} \partial_{k} = 0. Abstract purely algrbraic portion of this situation and consider any sequence of abelian groups A_{k} and homomorphisms \partial_{k} : A_{k} \to A_{k-1} such that \partial_{k-1} \partial_{k} = 0 for k \ge 1. Often A_{k} = 0 for k < 0 and k > n in applications.

(Def 44.1) Chain complex \langel A, \partial \rangle is a doubly infinite sequence A = { \cdots , A_{2} , A_{1}, A_{0}, A_{-1} , A_{-2}, \cdots } of abelian groups A_{k}, together with a collection \partial = {\partial_{k} | k \in \mathbb{Z} } of homomorphisms s.t. \partial_{k} : A_{k} \to A_{k-1} and \partial_{k-1} \partial_{k} = 0.

(Thm 44.2) If A is a chain complex, then the image under \partial_{k} is a subgroup of the kernel of \partial_{k-1}.

(Def 44.3) If A is a chain complex, then the kernel Z_{k} (A) of \partial_{k} is the group of k-cycles, and the image B_{k} (A) = \partial_{k+1} [A_{k+1}] is the group of k-boundaries. The factor group H_{k} (A) = Z_{k} (A) / B_{k} (A) is the kth homology group of A.

(Fundamental lemma) (Thm 44.4) Let A and A’ with collections \partial and \partial ‘ of homomorphisms be chain complexes, and suppose that there is a collection f of hoomorphisms f_{k} : A_{k} \to A’_{k} as indicated in the diagram .

Suppose, furthermore, that every square is commutative, that is, f_{k-1} \partial_{k} = \partial’{k} f{k} for all k, Then f_{k} induces a natural homomorphism f_{*k} : H_{k} (A) \to H_{k} (A’).

(Def) If the collections of maps f, \partial, and \partial’ have the property f_{k-1} \partial_{k} = \partial’_{k}, that the squares are commutative, then f commutes with \partial.

(Def 44.5) A chain complex \langle A’ , \partial’ \rangle is a subcomplex of a chain complex \langle A , \partial \rangle , if for all k, A_{k}’ is a subgroup of A_{k{ and \partial’{k} (C) = \partial{k} (c) for every c \in A’{k} , that is \partial’{k} and \partial_{k} have the same effect on elements of the subgroup A’{k} of A{k}.

(subsection) Relative Homology

(Def) Suppose that A’ is a subcomplex of the chain complex A. simplical subcomplex Y of a simplical complex X. We can then naturally consider C_{k} (Y) a subgroup of C_{k}(X). \partial_{k} [C_{k} (Y) ] \le C_{k-1} (Y)

If A’ is a subcomplex of the chain complex A, we can form the collection A/A’ of factor groups A_{k}/A’{k} . a collection \bar{\partial} of homomorphisms \bar{\partial{k} } : ( A_{k} / A’{k} ) \to (A{k-1} / A’{k-1}) : \bar{\partial{k}} (c + A’{k}) = \partial{k} (c) + A’{k-1} for c \in A{k}. then \bar{\partial}{k-1} \bar{\partial}{k} = 0

(Thm 44.7) If A’ is a subcomplex of the chain complex A, then the collection A/A’ of factor groups A_{k} / A’{k} together with collection \bar{\partial} of homomorphisms \bar{\partial}{k} defined by \bar{\partial_{k}} (c + A’{k}) = \partial{k} (c) + A’{k-1} for c \in A{k} is a chain complex.

(Def) The homology group H_{k} (A/A’) is the kth relative homology group of A modulo A’.

(subsection) Exact Homology Sequence of a Pair

(Lem 44.14) Let A’ be a subcomplex of a chain complex A. Let j be the collection of natural homomorphisms j_{k} : A_{k} \to (A_{k}/A’{k}). Then j{k-1} \parital_{k} = \bar{\partial}{k} j{k}, that is, j commutes with \partial.

(Thm 44.15) the map j_{k} of (Lem 44.14) induces a natural homomorphism j_{*k} : H_{k} (A) \to H_{k} (A/A’) .

(Lem 44.16) The map \partial_{*k} : H_{k} (A/A’) \to H_{k-1} (A’) : \partial_{*k}(h) = \partial_{k} (c) + B_{k-1} (A’) is well defined and is a homomorphism of H_{k} (A/A’) into H_{k-1} (A’).

(Lem 44.17) Let i_{*k} be the map i_{*k} : H_{k}(A’) \to H_{k}(A) induced from collection i of injection mappings i_{k} : A’{k} \to A{k} : i_{k}(c) = c for c \in A’_{k}. We can construct a following diagram.

The groups in diagram, together with given maps, form a chain complex.

(Def 44.18) A sequence of groups A_{k} and homomorphisms \parital_{k} forming a chain complex is an exact sequence if all the homology groups of the chain complex are 0, that is, if for all k we have that the image under \partial_{k} is equal to the kernel of \partial_{k-1}.

(Thm 44.19) The groups and maps of the chain complex in diagram (Lme 44.17) form an exact sequence.

(Def 44.20) The exact sequence in diagram (Lem 44.17) is the exact homology sequence of the pair (A,A’).

Factorization

(Section IX) Factorization

(Section 45) Unique Factorization Domains

(Def 45.1) Let R be a commutative ring with unity and let a, b \in R. If there exists c \in R s.t. b = ac, then a divides b (or a is a factor of b), denoted by a|b .

We read a \nmid b as “a does not divide b.”

(Def 45.2) an element u of a commutative ring with unity R is a unit of R if u divides 1, that is, if u has a multiplicative inverse in R. Two elements a, b \in R are associated in R if a = bu, where u is a unit in R.

(Def 45.4) A nonzero element p that is not a unit of an integral domain D is an irreducible of D if in every factorization p = ab in D has the property that either a or b is a unit.

(Def 45.5) An integral domain D is a unique factorization domain (UFD) if the following conditions are satisfied.

Every element of D that is neither 0 nor a unit can be factored into a product of a finite number of irreducibles.

If p_{1} \cdots p_{r} and q_{1} \cdots q_{s} are two factorizations of the same element of D into irreducibles, then r = s and the q_{j} can be renumbered so that p_{i} and q_{i} are associates.

(Def 45.7) An integral domain D is a principal ideal domain (PID) if every ideal in D is a principal ideal.

(subsection) Every PID is a UFD

(Def 45.8) If {A_{i} | i \in I} is a collection of sets, then the union \bigcup_{i \in I} A_{i} of the sets A_{i} is the set of all x s.t. x \in A_{i} for at least one i \in I.

(Lem 45.9) Let R be a commutative ring and let N_{1} \subseteq N_{2} \subseteq \cdots be an ascending chain of ideals N_{i} in R. Then N = \bigcup_{i} N_{i} is an ideal of R.

(Ascending Chain condition for a PID) (Lem 45.10) Let D be a PID. If N_{1} \subseteq N_{2} \subseteq \cdots is an ascending chain of ideals N_{i}, then there exists a positive integer r s.t. N_{r} = N_{s} for all s \ge r. Equivalently, every strictly ascending chain of ideals (all inclusions proper) in a PID is of finite length.

We express this by saying that the ascending chain condition (ACC) holds for ideals in a PID.

(Thm 45.11) Let D be a PID. Every element that is neither 0 nor a unit in D is a product of irreducibles.

(Lem 45.12) An ideal \langle p \rangle in a PID is maxial iff p is an irreducible.

(Lem 45.13) In a PID, if an irreducible p divides ab, then either p | a or p | b.

(Cor 45.14) If p is an irreducible in a PID and p divides the product a_{1}a_{2} \cdots a_{n} for a_{i} \in D , then p | a_{i} for at least one i.

(Def 45.15) A nonzero nonunit element p of an integral domain D is a prime if, for all a, b \in D, p | ab implies either p | a or p | b.

(Thm 45.17) Every PID is a UFD.

(Fundamental Theorem of Arithmetic) (Cor 45.18) The integral domain \mathbb{Z} is a UFD.

(subsection) If D is a UFD, then D[x] is a UFD

(Def 45.19) Let D be a UFD and let a_{1}, a_{2}, \cdots , a_{n} be nonzero elements of D. An element d of D is a greatest common divisor (gcd) of all of the a_{i} if d | a_{i} for i = 1, \cdots ,n and any other d’ \in D that divides all the a_{i} also divides d.

(Def 45.21) Let D be a UFD. a nonconstant polynomial f(x) = a_{0} + a_{1} x + \cdots + a_{n} x^{n} in D[x] is primitive if 1 is a gcd of the a_{i} for i = 0,1, …, n.

(Lem 45.23) If D is a UFD, then for every nonconstant f(x) \in D[x] we have f(x) = (c)g(x), where c \in D, g(x) \in D[x], and g(x) is primitive. The element c is unique up to a unit factor in D and is the content of f(x). Also g(x) is unique up to a unit factor in D.

(Gauss’s Lemma) (Lem 45.25) If D is a UFD, then a product of two primitive polynomials in D[x] is again primitive.

(Cor 45.26) If D is a UFD, then a finite product of primitive polynomials in D[x] is again primitive.

(Lem 45.27) Let D be a UFD and let F be a field of quotients of D. Let f(x) \in D[x], where (degree f(x)) > 0. If f(x) is an irreducible in D[x], then f(x) is also an irreducible in F[x]. Also, if f(x) is primitive in D[x] and irreducible in F[x], then f(x) is irreducible in D[x].

(Cor 45.28) If D is a UFD and F is a field of quotients of D, then a nonconstant f(x) \in D[x] factors into a product of two polynomials of lower degrees r and s in F[x] iff it has a factorization into polynomials of the same degrees r and s in D[x].

(Thm 45.29) If D is a UFD, then D[x] is a UFD.

(Cor 45.30) If F is a field and x_{1} , \cdots x_{n} are indeterminates, then F[x_{1}, \cdots x_{n} ] is a UFD.

Euclidean Domains

(Section 46) Euclidean Domains

(Def 46.1) Euclidean norm on an integral domain D is a function \nu mapping the nonzero elements of D into the nonnegative integers s.t. the following conditions are satisfied.

For all a,b \in D with b \neq 0, there exist q and r in D s.t. a = bq + r, where either r = 0 or \nu (r) < \nu (b).

For all a, b \in D, where neither a nor b is 0, \nu (a) \le \nu (ab).

An integral domain D is a Euclidean domain if there exists a Euclidean norm on D.

(Thm 46.4) Every Euclidean domain is a PID.

(Cor 46.5) A Euclidean domain is a UFD.

(subsection) Arithmetic in Euclidean Domains

(Thm 46.6) For a Euclidean domain with a Euclidean norm \nu, \nu(1) is minimal among all \nu (a) for nonzero a \in D, and u \in D is a unit iff \nu (u) = \nu (1).

(Euclidean Algorithm) (Thm 46.9) Let D be a Euclidean domain with a Euclidean norm \nu, and let a and b be nonzero elements of D. Let r_{1} be as in first condition for a Euclidean norm, that is a = bq_{1} + r_{1} where either r_{1} = 0 or \nu (r_{1}) < \nu (b) . If r_{1} \neq 0, let r_{2} be s.t. b = r_{1} q_{2} + r_{2} where either r_{2} = 0 or \nu (r_{2}) < \nu (r_{1}) . In general, let r_{i+1} be s.t. r_{i-1} = r_{i} q_{i+1} + r_{i+1} where either r_{i+1} = 0 or \nu (r_{i+1}) < \nu (r_{i}) . Then the sequence r_{i}, r_{2}, \cdots must terminate with some r_{s} = 0. If r_{1} = 0, then b is a gcd of a and b. If r_{1} \neq 0 and r_{s} is the first r_{i} = 0, then a gcd of a and b is r_{s-1}. Furthemore, if d is a gcd of a and b, then there exist \lambda and \mu in D s.t. d = \lambda a + \mu b.

Automorphisms and Galois Theory

(Section X) Automorphisms and Galois Theory

(Section 48) Automorphisms of Fields.

(subsection) The conjugation isomorphisms of Algebraic Field Theory

(Def) all algebraic extensions and all elements algebraic over a field F under consideration are contained in one fixed algebraic closure \bar{F} of F.

(Def 48.1) Let E be an algebraic extension of a field F. Two elements \alpha , \beta \in E are conjugate over F if irr( \alpha , F ) = irr ( \beta , F ) , that is, if \alpha and \beta are zeros of the same irreducible polynomial over F.

(The Conjugation Isomorphisms) (Thm 48.3) Let F be a field, and let \alpha and \beta be algebraic over F with deg ( \alpha , ) = n. The map \psi_{\alpha , \beta } : F( \alpha ) \to F( \beta ) defined by \psi_{\alpha , \beta } (c_{0} + c_{i} \alpha + \cdots + c_{n-1} \alpha^{n-1} ) = c_{0} + c_{1} \beta + \cdots + c_{n-1} \beta^{n-1} for c_{i} \in F is an isomorphism of F( \alpha ) onto F( \beta ) iff \alpha and \beta are conjugate over F.

(Cor 48.5) Let \alpha be algebraic over a field F. Every isomorphism \psi mapping F( \alpha ) onto a subfield of \bar{F} s.t. \psi (a) = a for a \in F maps \alpha onto a conjugate \beta of \alpha over F. Conversely, for each conjugate \beta of \alpha over F, there exists exactly one isomorphism \psi_{\alpha , \beta } of F( \alpha ) onto a subfield of \bar{F} mapping \alpha onto \beta and mapping each a \in F onto itself.

(Cor 48.6) Let f(x) \in \mathbb{R} [x] . If f(a + bi) = 0 for (a + bi) \in \mathbb{C}, where a, b \in \mathbb{R}, then f(a – bi) = 0 also. Loosely, complex zeros of polynomials with real coefficient occur in conjugate pairs.

(subsection) Automorphisms and Fixed Fields

(Def 48.8) An isomorphism of a field into itself is an automorphism of the field.

(Def 48.9) If \sigma is an isomorphism of a field E onto some field, then an element a of E is left fixed by \sigma if \sigma (a) = a. A collection S of isomorphisms of E leaves a subfield F of E fixed if each a \in F is left fixed by every \sigma \in S. If { \sigma } leaves F fixed, then \sigma leaves F fixed.

(Thm 48.11) Let \( \lbrace \sigma_{i} \vert i \in I \rbrace \) be a collection of automorphisms of a field E. Then the set \(E_{\lbrace \sigma_{i} \rbrace }\) of all a \in E left fixed by every \sigma_{i} for i \in I forms a subfield of E.

(Def 48.12) The field \(E_{\lbrace \sigma_{i} \rbrace }\) of (Thm 48.11) is the fixed field of \( \lbrace \sigma_{i} \vert i \in I \rbrace \) . For a single automorphism \sigma, we shall refer to E_{ {\sigma } } as the fixed field of \sigma.

(Thm 48.14) The set of all automorphisms of a field E is a group under function composition.

(Thm 48.15) Let E be a field, and let F be a subfield of E, Then the set G( E/F) of all automorphisms of E leaving F fixed forms a subgroup of the group of all automorphisms of E. Furthermore, F \le E_{ G(E/F) } .

(Def 48.16) The group G(E/F) of the preceding theorem is the group of automorphisms of E leaving F fixed, or more briefly, the group of E over F.

(subsection) The Frobenius Automorphism

(Thm 48.19) Let F be a finite field of characteristic p. Then the map \sigma_{p} : F \to F defined by \sigma_{p} (a) = a^{p} for a \in F is an automorphism, the Frobenius automorphism, of F. Also, F_{ \sigma_{p} } \simeq \mathbb{Z}_{p} .

The Isomorphism Extension Theorem

(Section 49) The Isomorphism Extension Theorem

(subsection) Extension theorem

(Isomorphism Extension Theorem) (Thm 49.3) Let E be an algrbraic extension of a field F. Let \sigma be an isomorphism of F onto a field F’. Let \bar{F’} be an algebraic closure of F’. Then \sigma can be extended to an isomorphism \tau of E onto a subfield of \bar{F’} s.t. \tau(a) = \sigma(a) for all a \in F.

(Cor 49.4) If E \le \bar{F} is an algebraic extension of F and \alpha , \beta \in E are conjugate over F, then the conjugation isomorphism \psi_{\alpha , \beta } : F(\alpha ) \to F(\beta ) , given by (Thm 48.3) , can be extended to an isomorphism of E onto a subfield of \bar{F}.

(Cor 49.5) Let \bar{F} and \bar{F’} be two algebraic closures of F. Then \bar{F} is isomorphic to \bar{F’} under an isomorphism leaving each element of F fixed.

(subsection) Index of a Field Extension

(Thm 49.7) Let E be a finite extension of a field F. let \sigma be an isomorphism of F onto a field F’ , and let \bar{F’} be an algebraic closure of F’. Then the number of extensions of \sigma to an isomorphism \tau of E onto a subfield of \bar{F’} is finite, and independent of F’, \bar{F’}, and \sigma. That is, the number of extensions is completely determined by the two fields E and F; it is intrinsic to them.

(Def 49.9) Let E be a finite extension of a field F. The number of isomorphisms of E onto a subfield of \bar{F} leaving F fixed is the index [E : F] of E over F.

(Cor 49.10) If F \le E \le K, where K is a finite extension field of the field F, then [K : F] = [K : E] [E : F] .

Splitting Fields

(section 50) Splitting Fields

(Def 50.1) Let F be a field with algebraic closure \bar{F} . Let {f_{i} (x) | i \in I } be a collection of polynomials in F[x] . A field E \le \bar{F} is the splitting field of {f_{i} (X) |i \in I} over F if E is the smallest subfield of \bar{F} containing F and all the zeros in \bar{F} of each of the f_{i} (x) for i \in I . A field K \le \bar{F} is a splitting field over F if it is the splitting field of some set of polynomials in F[x].

(Thm 50.3) A field E, where F \le E \le \bar{F}, is a splitting field over F iff every automorphism of \bar{F} leaving F fixed maps E onto itself and thus induces an automorphism of E leaving F fixed.

(Def 50.4) Let E be an extension field of a field F. A polynomial f(x) \in F[x] splits in E if it factors into a product of linear factors in E[x].

(Cor 50.6) If E \le \bar{F} is a splitting field over F ,then every irreducible polynomial in F[x] having a zero in E splits in E.

(Cor 50.7) If E \le \bar{F} is a splitting field over F, then every isomorphic mapping of E onto a subfield of \bar{F} and leaving F fixed is actually an automorphism of E. In particular, if E is a splitting field of finite degree over F, then {E : F} = |G(E/F)|.

Separable Extensions

(Section 51) Separable Extensions

(subsection) Multiplicity of Zeros of a Polynomial

(Def 51.1) Let f(x) \in F[x]. An element \alpha of \bar{F} s.t. f( \alpha ) = 0 is a zero of f(x) of multiplicity \nu if \nu is the greatest integer s.t. (x- \alpha )^{\nu} is a factor of f(x) in \bar{F} [x].

(Thm 51.2) Let f(x) be irreducible in F[x]. Then all zeros of f(x) in \bar{F} have the same multiplicity.

(Cor 51.3) If f(x) is irreducible in F[x], then f(x) has a factorization in F[x] of the form a \prod_{i} (x- \alpha_{i} )^{\nu} , where the \alpha_{i} are the distinct zeros of f(x) in \bar{F} and a \in F.

(prop) {F(\alpha ) : F } is the number of distinct zeros of irr( \alpha , F).

(Thm 51.6) If E is a finite extension of F, then {E : F} divides [E : F].

(subsection) Separable Extensions

(Def 51.7) A finite Extension E of F is a separable extension of F if {E : F} = [E : F] . An element \alpha of \bar{F} is separable over F if F( \alpha ) is a separable extension of F. An irreducible polynomial f(x) \in F[x] is separable over F if every zero of f(x) in \bar{F} is separable over F.

(Thm 51.9) If K is a finite extension of E and E is a finite extension of F , that is F \le E \le K, then K is separable over F iff K is separable over E and E is separable over F.

(Cor 51.10) If E is a finite extension of F, then E is separable over F iff each \alpha in E is separable over F.

(subsection) Perfect Fields

(Lem 51.11) Let \bar{F} be an algebraic closure of F, and let f(x) = x^{n} + a_{n-1} x^{n-1} + \cdots + a_{1} x + a_{0} be any monic polynomial in \bar{F} [x] . If (f(x))^{m} \in F[x] and m \cdot 1 \neq 0 in F, then f(x) \in F[x] , that is, all a_{i} \in F.

(Def 51.12) A field is perfect if every finite extension is a separable extension.

(Thm 51.13) Every field of characteristic zero is perfect.

(Thm 51.14) Every finite field is perfect.

(subsection) Primitive element theorem

(Primitive element theorem) (Thm 51.15) Let E be a finite separable extension of a field F. Then there exists \alpha \in E s.t. F = F(\alpha) . (Such an element \alpha is a primitive element.)

That is, a finite separable extension of a field is a simple extension.

(Cor 51.16) A finite extension of a field of characteristic zero is a simple extension.

Totally Inseparable Extensions

(Section 52) Totally Inseparable Extensions

(Def 52.1) A finite extension E of a field F is a totally inseparable extension of F if {E : F} = 1 < [E : F] . an element \alpha of \bar{F} is totally inseparable over F if F(\alpha) is totally inseparable over F.

(Thm 52.3) If K is a finite extension of E, E is a finite extension of F, and F < E < K, then K is totally inseparable over F iff K is totally inseparable over E and E is totally inseparable over F.

(Cor 52.4) If E is a finite extension of F, then E is totally inseparable over F iff each \alpha in E, \alpha \neq F, is totally inseparable over F.

(subsection) Separable Closures

(Thm 52.5) Let F have characteristic p \neq 0, and let E be a finite extension of F. Then \alpha \in E, \alpha \notin F, is totally inseparable over F iff there is sime integer t \ge 1 s.t. \alpha^{p^{i}} \in F.

Furthermore, there is a unique extension K of F, with F \le K \le E, s.t. K is separable over F, and either E = K or E is totally inseparable over K.

(Def 52.6) The unique field K of (Thm 52.5) is the separable closure of F in E.

Galois Theory

(Section 53) Galois Theory

(Recall)

Let F \le E \le \bar{F}, \alpha \in E , and let \beta be a conjugate of \alpha over F, irr(\alpha F) has \beta as a zero also. Then there is an isomorphism \psi_{\alpha , \beta } mapping F(\alpha ) onto F(\beta ) that leaves F fixed and maps \alpha onto \beta.

If F \le E \le \bar{F} and \alpha \in E, then an automorphism \sigma of \bar{F} that leaves F fixed must map \alpha onto some conjugate of \alpha over F.

If F \le E, the collection of all automotphisms of E leaving F fixed forms a group G(E/F) . For any subset S of G(E/F) , the set of all elements of E left fixed by all elements of S is a field E_{s} . Also, F \le E_{G(E/F)} .

A field E , F \le E \le \bar{F}, is a splitting field over F iff every isomorphism of E onto a subfield of \bar{F} leaving F fixed is an automorphism of E. If E is a finite extension and a splitting field over F, then |G(E/F)| = {E : F} .

If E is a finite extension of F, then {E : F} divides [E : F] . If E is also separable over F, then {E : F} = [E : F]. Also, E is separable over F iff irr( \alpha , F ) has all zeros of multiplicity 1 for every \alpha \in E.

If E is a finite extension of F and is separable splitting field over F, then |G(E/F)| = {E:F} = [E : F].

(subsection) Normal Extensions

(Def 53.1) A finite extension K of F is a finite normal extension of F if K is a separable splitting field over F.

(Thm 53.2) Let K be a finite normal extension of F, and let E be an extension of F, where F \le E \le K \le \bar{F}. Then K is a finite normal extension of E< and G(K/E) is precisely the subgroup of G(K/F) consisting of all those automorphisms that leave E fixed.

Moreover, two automorphisms \sigma and \tau in G(K/F) induce the same isomorphisms of E onto a subfield of \bar{F} iff they are in the same left coset of G(K/E) in G(K/F).

(subsection) Main Theorem

(Def 53.5) If K is a finite normal extension of a field F, then G(K/F) is the Galois group of K over F.

(Main theorem of Galois Theory) (Thm 53.6) Let K be a finite normal extension of a field F, with Galois group G(K/F) . For a field E< where F \le E \le K, let \lambda(E) be the subgroup of G(K/F) leaving E fixed. Then \lambda is a one-to-one map of the set of all such immediate fields E onto the set of all subgroups of G(K/F). The following properties hold for \lambda.

\lambda (E) = G(K/E)

E = K_{G(K/E)} = K_{\lambda (E)}

For H \le G(K/F) , \lambda(E_{H}) = H

[K : E] = |\lambda (E)} and [E : F] = (G(K/F) : \lambda(E) ), the number of left cosets of \lambda(E) in G(K/F).

E is a normal extension of F iff \lambda(E) is a normal subgroup of G(K/F). When \lambda(E) is a normal subgroup of G(K/F), then G(E/F) \simeq G(K/F) / G(K/E).

The diagram of subgroups of G(K/F) is the inverted diagram of intermediate field of K over F.

(prop) The Galois group G(K/F) is the group of polynomial f(x) over F.

(subsection) Galois Groups over Finite fields

(Thm 53.6) Let K be a finite extension of degree n of a finite field F of p^{r} elements. Then G(K/F) is cyclic of order n, and is generated by \sigam_{p^{r}} , where for \alpha \in K, \sigma_{p^{r}} (\alpha) = \alpha^{p^{r}}.

Illustrations of Galois Theory

(Section 54) Illustrations of Galois Theory

(subsection) Symmetric Functions

(Def) Let F be a field, and let y_{1}, \cdots , y_{n} be indeterminates. There are some natural automorphisms of F(y_{1}, \cdots , y_{n}) leaving F fixed, those defined by permutations of {y_{1}, \cdots , y_{n} . Let \sigma be a permutation of {1, \cdots , n}, that is, \sigma \in S_{n}. Then \sigma gives rise to a natural map \bar{\sigma} : F(y_{1}, \cdots , y_{n}) \to F(y_{1}, \cdots , y_{n} ) given by \bar{\sigma} \biggl( \frac{f(y_{1} , \cdots, y_{n} ) }{ g(y_{1}, \cdots, y_{n}) } \biggr) = \frac{f(y_{\sigma (1)} , \cdots, y_{\sigma (n)} ) }{ g(y_{\sigma (1)}, \cdots, y_{\sigma (n)})} for f(y_{1}, \cdots, y_{n}) , g(y_{1}, \cdots, y_{n}) \in F[y_{1}, \cdots , y_{n} ] , with g(y_{1}, \cdots, y_{n}) \neq 0. \bar{sigma} is an automorphism of F(y_{1}, \cdots, y_{n}) leaving F fixed.

The elements F(y_{1}, \cdots, y_{n}) left fixed by all \bar{\sigma} , for all \sigma \in S_{n}, are those rational functions that are symmetric in the indeterminates y_{1}, \cdots , y_{n} .

(Def 54.1) An element of the field F(y_{1}, \cdots , y_{n}) is a symmetric function in y_{1}, \cdots , y_{n} over F, if it is left fixed by all permutations of y_{1}, \cdots , y_{n}, in the sense as explained above.

(Thm 54.2) Let s_{1}, \cdots , s_{n} be the elementary symmetric functions in the indeteminates y_{1}, \cdots , y_{n} . Then every symmetric function of y_{1}, \cdots, y_{n} over F is a rational function of the elementary symmetric funcitons,

Also, F(y_{1}, \cdots, y_{n} ) is a finite normal extension of degree n! of F(s_{1}, \cdots , s_{n}) , and the Galois group of this extension is naturally isomorphic to S_{n}.

Cyclotomic Extensions

(Section 55) Cyclotomic Extensions

(subsection) Galois group of a Cyclotomic Extension

(Def 55.1) The splitting field of x^{n} -1 over F is the nth cyclotomic extension of F.

(Def 55.2) The polynomial \Phi_{n} (x) = \prod_{i = 1}^{\phi (n)} (x- \alpha_{i} ) where the \alpha_{i} are the primitive nth roots of unity in \bar{F}, is the nth cyclotomic polynomial over F.

(prop) Over \mathbb{Q}, \Phi_{n}(x) is irreducible.

(Thm 55.4) The Galois group of the nth cyclotomic extension of \mathbb{Q} has \phi (n) elements and is isomorphic to the group consisting of the positive integers less than n and relatively prime to n under multiplication modulo n.

(Cor 55.6) The Galois group of the pth cyclotomic extension of \mathbb{Q} for a prime p is cyclic of order p-1.

(subsection) Constructible Polygons

(Def) Fermat prime p = 2^{2^{k}} + 1 for k \in \mathbb{N} which is prime.

(Thm 55.8) The regular n-gon is constructible with a compass and a straightedge iff all the odd primes dividing n are Fermat primes whose squares do not divide n.

Insolvability of the Quintic

(Section 56) Insolvability of the Quintic

(subsection) Extensions by Radicals

(Def 56.1) An extension K of a field F is an extension of F by radicals if there are elements \alpha_{1} , \cdots, \alpha_{r} \in K and positive integers n_{1}, \cdots, n_{r} s.t. K = F(\alpha_{1}, \cdots, \alpha_{r}) , \alpha_{1}^{n_{1}} \in F and \alpha_{i}^{n_{i}} \in F(\alpha_{1}, \cdots, \alpha_{i-1}) for 1 < i \le r. A polynomial f(x) \in F[x] is solvable by radicals over F if the splitting field E of f(x) over F is contained in an extension of F by radicals.

(Lem 56.3) Let F be a field of characteristic 0, and let a \in F . If K is the splitting field of x^{n} -a over F, then G(K/F) is a solvable group.

(Thm 56.4) Let F be a field of characteristic zero, and let F \le E \le K \le \bar{F}, where E is a normal extension of F and K is an extension of F by radicals. Then G(E/F) is a solvable group.

(subsection) Insolvability of the Quintic

(Def) Let y_{1} \in \mathbb{R} be transcendental over \mathbb{Q}, y_{1} \in \mathbb{R} be transcendental over \mathbb{Q}(y_{1}) , and so on, until we ger y_{5} \in \mathbb{R} transcendental over \mathbb{Q} (y_{1}, \cdots , y_{4} ). It can be shown by a counting argument that such transcendental real numbers exist. Transcendentals found in this fashion are independent transcendental elements over \mathbb{Q}.

(Def) Elementary symmetry functions in the y_{i}, namely

s_{1} = y_{1} + y_{2} + \cdots + y_{5}

s_{2} = y_{1}y_{2} + y_{1}y_{3} + y_{1}y_{4} + y_{1}y_{5} + y_{2}y_{3} + \cdots + y_{3}y_{5} + y_{4}y_{5} ,

\vdots

s_{5} = y_{1}y_{2}y_{3}y_{4}y_{5}

(Thm 56.6) Let y_{1}, \cdots, y_{5} be independent transcendental real numbers over \mathbb{Q}. The polynomial f(x) = \prod_{i = 1}^{5} (x-y_{i}) is not solvable by radicals over F = \mathbb{Q} (s_{1}, \cdots, s_{5}) , where s_{i} is the ith elementary symmetric function in y_{1}, \cdots , y_{5}.

Complex Analysis

Preliminaries to Complex Analysis

(Section 1) Preliminaries to Complex Analysis

(subsection 1) Complex numbers and the complex plane

(subsubsection) Basic Properties

(Def) A complex number takes the form z = x + iy where x and y are real. And i is an imaginary number that satisfies i^{2} = -1. We call x and y the real part and the imaginary part of z, and we write x = Re(z) and y = Im(z). A complex number with zero real part is said to be purely imaginary.

The set of all complex numbers is denoted by \mathbb{C}. The complex numbers can be visualized as the usual Euclidean plane : The complex number z = x + iy \in \mathbb{C} is identified with the point (x,y) \in \mathbb{R}^{2} .

x and y axis of \mathbb{R}^{2} are called the real axis and the imaginary axis.

(Def) If z_{1} = x_{1} + iy_{1} and z_{2} = x_{2} + iy_{2} , then z_{1} + z_{2} = (x_{1} + x_{2}) + i(y_{1} + y_{2}) and z_{1}z_{2} = (x_{1}x_{2} – y_{1}y_{2}) + i (x_{1}y_{2} + y_{1}x_{2}) .

Commutativity : z_{1} + z_{2} = z_{2} + z_{1} and z_{1}z_{2} = z_{2}z_{1} for all z_{1} , z_{2} \in \mathbb{C}.

Associativity : (z_{1} + z_{2}) + z_{3} = z_{1} + (z_{2} + z_{3}) ; and (z_{1}z_{2}) z_{3} = z_{1}(z_{2}z_{3}) for z_{1},z_{2},z_{3} \in \mathbb{C}.

Distributivity : z_{1}(z_{2}+z_{3}) = z_{1}z_{2} + z_{1}z_{3} for all z_{1}, z_{2}, z_{3} \in \mathbb{C}.

(Def) The absolute value of a complex number z = x + iy by |z| = (x^{2} + y^{2})^{1/2} , so that |z| is precisely the distance from the origin to the point (x,y). The triangle inequality |z+w| \le |z| + |w| for all z, w \in \mathbb{C}.

For all z \in \mathbb{C} we have both |Re(z)| \le |z| and |Im(z)| \le |z| , and for all z,w \in \mathbb{C}, ||z|-|w|| \le |z-w| .

The complex conjugate of z = x + iy is defined by \bar{z} = x – iy.

(prop) z is real iff z = \bar{z} and purely imaginary iff z = -\bar{z}.

Re(z) = \frac{z+\bar{z}}{2} and Im(z) = \frac{z-\bar{z}}{2i}.

\frac{1}{z} = \frac{\bar{z}}{|z|^{2}} whenever z \neq 0.

(Def) Polar form z = re^{i\theta} where r >0. Also \theta \in \mathbb{R} is called the argument of z and denoted by arg z.

e^{i\theta} = cos \theta + i sin \theta.

If z = re^{i\theta} and w = se^{i\phi} , then zw = rs e^{i(\theta + \phi)}.

(subsubsection 1.2) Convergence

(Def) A sequence {z_{1}, z_{2}, …} of complex numbers is said to converge to w \in \mathbb{C} if \lim_{n \to \infty| |z_{n} – w| = 0 , and we write w = \lim_{n \to \infty} z_{n}.

A sequence {z_{n}} is said to be a Cauchy sequence (or simply Cauchy) if |z_{n} – z_{m}| \to 0 as n , m \to \infty. In other words, given \epsilon >0 there exists an integer N>0 so that |z_{n} – z_{m}| < \epsilon whenever n,m >N.

(Prop) \mathbb{R} is complete : every Cauchy sequence of real numbers converges to a real number. The sequence {z_{n}} is Cauchy iff the sequences of real and imaginary parts of z_{n} are.

(Thm 1.1) \mathbb{C} , the complex numbers, is complete.

(subsubsection 1.3.) Sets in the complex plane

If z_{0} \in \mathbb{C} and r > 0 , we define the open disc D_{r}(z_{0}) of radius r centered at z_{0} to be the set of all complex numbers that are at absolute value strictly less than r from z_{0}. D_{r} (z_{0}) = {z \in \mathbb{C} : |z – z_{0}| < r}.

The closed disc \bar{D}{r}(z{0}) = {z \in \mathbb{C} : |z – z_{0}| < r} and the boundary of either the open or closed disc is the circle C_{r}(z_{0}) = {z \in \mathbb{C} : |z-z_{0}| = r.

Denote unit disc \mathbb{D} = {z \in \mathbb{C} : |z|<1}.

Given a set \Omega \subset \mathbb{C}, a point z_{0} is an interior point of \Omega if there exists r>0 s.t. D_{r}(z_{0}) \subset \Omega.

The interior of \Omega consists of all its interior points. A set \Omega is open if every point in that set is an interior point of \Omega. A set \Omega is closed if its complement \Omega^{c} = \mathbb{C} - \Omega is open. A point z \in \mathbb{C} is said to be a limit point of the set \Omega if there exists a sequence of points z_{n} \in \Omega s.t. z_{n} \neq z and \lim_{n \to \infty} z_{n} = z. The closure of any set \Omega is the union of \Omega and its limit points, and is often denoted by \bar{\Omega}.

Boundary of a set \Omega is equal to its closure minus its interior, and is often denoted by \partial \Omega. A set \Omega is bounded if there exists M>0 s.t. |z|<M whenever z \in \Omega. If \Omega is bounded, we define its diameter by diam(\Omega) = \sup_{z,w \in \Omega} |z-w|.

A set \Omega is said to be compact if it is closed and bounded.

(Thm 1.2.) The set \Omega \subset \mathbb{C} is compact iff every sequence {z_{n}} \subset \Omega has a subsequence that converges to a point in \Omega.

(Def) An open covering of \Omega is a family of open sets {U_{\alpha}} (not necessarily countable) s.t. \Omega \subset \bigcup_{\alpha} U_{\alpha}.

(Thm 1.3.) A set \Omega is compact iff every open covering of \Omega has a finite subcovering.

(prop 1.4.) If \Omega_{1} \supset \Omega_{2} \supset \cdots \supset \Omega_{n} \supset \cdots is a sequence of non-empty compact sets in \mathbb{C} with the property that diam(\Omega_{n}) \to 0 as n \to \infty, then there exists a unique point w \in \mathbb{C} s.t. w \in \Omega_{n} for all n.

(Def) An open set \Omega \subset \mathbb{C} is said to be connected if it is not possible to find two disjoint non-empty open sets \Omega_{1} and \Omega_{2} s.t. \Omega = \Omega_{1} \cap \Omega_{2}.

A connected open set in \mathbb{C} will be called a region. Similarly, a closed set F is connected if one cannot write F = F_{1} \cap F_{2} where F_{1} and F_{2} are disjoint non-empty closed sets.

(subsection 2) Functions on the complex plane

(subsubsection 2.1.) Continuous functions

(Def) Let f be a function defined on a set \Omega of complex numbers. We say that f is continuous at the point z_{0} \in \Omega if for every \epsilon >0 there exists \delta>0 s.t. whenever z \in \Omega and |z-z_{0}| < \delta then |f(z) – f(z_{0}) | < \epsilon.

(Prop) The function f is said to be continuous on \Omega if it is continuous at every point of \Omega. Sums and products of continuous functions are also continuous.

The function f of the complex argument z = x + iy is continuous iff it is continuous viewed as a function of the two real variables x and y.

(Def) f attains a maximum at the point z_{0} \in \Omega if |f(z)| \le |f(z_{0})| for all z \in \Omega, with the inequality reversed for the definition of a minimum.

(Thm 2.1.) A continuous function on a compact set \Omega is bounded and attains a maximum and minimum on \Omega.

(subsubsection 2.2.) Holomorphic functions

(Def) Let \Omega be an open set in \mathbb{C} and f a complex-valued function on \Omega. The function f is holomorphic at the point z_{0} \in \Omega if the quotient \frac{f(z_{0} + h) – f(z_{0})}{h} converges to a limit when h \to 0. The limit of the quotient, when it exists, is denoted by f’(z_{0}) , and is called the derivative of f at z_{0} ; f’(z_{0}) = \lim_{h \to 0} \frac{f(z_{0} + h) – f(z_{0})}{h}

(Def) The function f is said to be holomorphic on \Omega if f is holomorphic at every point of \Omega. If C is a closed subset of \mathbb{C}, we say that f is holomorphic on C if f is holomorphic in some open set containing C . Finally, if f is holomorphic in all of \mathbb{C} we say that f is entire.

The terms regular or complex differentiable or analytic are used instead of holomorphic occasionally.

(Prop) A function f is holomorphic at z_{0} \in \Omega iff there exists a complex number a s.t. f(z_{0} + h) – f(z_{0}) – ah = h \psi(h) , where \psi is a function defined for all small h and \lim_{h \to 0} \psi(h) = 0. f is continuous whenever it is holomorphic.

(Prop 2.2.) If f and g are holomorphic in \Omega, then

f + g is holomorphic in \Omega and (f + g)’ = f’ + g’.

fg is holomorphic in \Omega and (fg)’ = f’g + fg’.

If g(z_{0}) \neq 0, then f/g is holomorphic at z_{0} and (f/g)’ = \frac{f’g – fg’}{g^{2}}.

Moreover, if f : \Omega \to U and g : U \to \mathbb{C} are holomorphic, the chain rule holds

(g \bullet f)’(z) = g’(f(z))f’(z) for all z \in \Omega.

(Def) Let F(x,y) = (u(x,y), v(x,y)) . If F is differentiable, the partial derivative of u and v exist, and the linear transformation J is described in the standard basis of \mathbb{R}^{2} by the Jacobian matrix of F J = J_{F}(x,y) = \begin{pmatrix} \frac{\partial u}{\partial x} & \frac{\partial u}{\partial y } \ \frac{\partial v}{\partial x} & \frac{\partial v}{\partial y}\end{pmatrix}

(Def) \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} and \frac{\partial u}{\partial y} = - \frac{\partial v}{\partial x} These are called Cauchy-Riemann equations.

(Def) \frac{\partial}{\partial z} = \frac{1}{2} (\frac{\partial}{\partial x} + \frac{1}{i} \frac{\partial}{\partial y} ) and \frac{\partial}{\partial \bar{z}} = \frac{1}{2} (\frac{\partial}{\partial x} - \frac{1}{i} \frac{\partial}{\partial y} )

(Prop 2.3.) If f is holomorphic at z_{0}, then \frac{\partial f}{\partial \bar{z}} (z_{0}) = 0 and f’(z_{0}) = \frac{\partial f}{\partial z} (z_{0}) = 2 \frac{\partial u}{\partial z} (z_{0}) .

Also, if we write F(x,y) = f(z), then F is differentiable in the sense of real variables, and det J_{F} (x_{0}, y_{0}) = |f’(z_{0})|^{2}.

(Thm 2.4.) Suppose f = u + iv is a complex-valued function defined on an open set \Omega. If u and v are continuously differentiable and satisfy the Cauchy-Riemann equations on \Omega, then f is holomorphic on \Omega and f’(z) = \partial f / \partial z.

(subsection 2.3.) Power series

(Def) Complex exponential function which is defined for z \in \mathbb{C} by e^{z} = \sum_{n = 0}^{\infty} \frac{z^{n}}{n!} . This series converges absolutely for every z \in \mathbb{C}.

e^{z} is holomorphic in all of \mathbb{C}. e^{z} is its on derivative.

(Def) A power series is an expansion of the form \sum_{n = 0}^{\infty} a_{n} z^{n}.

(Thm 2.5.) Given a power series \sum_{n = 0}^{\infty} a_{n} z^{n}, there exists 0 \le R \le \infty s.t.

If |z| < R the series converges absolutely.

If |z| > R the series diverges.

Moreover, if we use the convention that 1/0 = \infty and 1/\infty = 0, then R is given by Hadamard’s formula 1/R = \limsum |a_{n}|^{1/n}.

The number R is called the radius of convergence of the power series, and the region |z|<R the disc of convergence. In particular, we have R = \infty in the case of exponential function, and R = 1 for the geometric series.

(Def) Trigonometric functions : cos z = \sum_{n = 0}^{\infty} (-1)^{n} \frac{z^{2n}}{(2n)!} and sin z = \sum_{n = 0}^{\infty} (-1)^{n} \frac{z^{2n+1}}{(2n+1)!}.

(Prop) cos z = \frac{e^{iz} + e^{-iz}}{2} and sin z = \frac{e^{iz} – e^{-iz}}{2i} These are called the Euler formulas for the cosine and sine functions.

(Thm 2.6.) The power series f(z) = \sum_{n = 0}^{\infty} a_{n}z^{n} defines a holomorphic function in its disc of convergence. The derivative of f is also a power series obtained by differentiating term by term the series for f, that is, f’(z) = \sum_{n = 0}^{\infty} nan_{n} z^{n-1}.

Moreover, f’ has the same radius of convergence as f.

(Cor 2.7.) A power series is infinitely complex differentiable in its disc of convergence, and the higher derivatives are also power series obtained by termwise differentiation.

(Def) A power series centered at z_{0} \in \mathbb{C} is an expression of the form f(z) = \sum_{n = 0}^{\infty} a_{n}(z – z_{0})^{n}. The disc of convergence of f is now centered at z_{0} and its radius is still given by Hadamard’s formula.

(Def) A function f defined on an open set \Omega is said to be analytic (or have a power series expansion) at a point z_{0} \in \Omega if there exists a power series \sum a_{n} (z-z_{0})^{n} centered at z_{0}, with positive radius of convergence, s.t. f(z) = \sum_{n = 0}^{\infty} a_{n}(z-z_{0})^{n} for all z in a neighborhood of z_{0}.

If f has a power series expansion at every point in \Omega, we say that f is analytic on \Omega.

(subsection 3) Integration along curves

(Def) A parametrized curve is a function z(t) which maps a closed interval [a,b] \subset \mathbb{R} to the complex plane.

(Def) We say that the parametrized curve is smooth if z’(t) exists and is continuous at [a,b] , and z’(t) \neq 0 for t \in [a,b] . At the points t = a and t = b, the quantities z’(a) and z’(b) are interpreted as the one-sided limits z’(a) = \lim_{h \to 0 , h >0} \frac{z(a+h) – z(a)}{h} and z’(b) = \lim_{h \to 0, h < 0} \frac{z(b+h) – z(b)}{h}

(Def) We say that the parametrized curve is piecewise-smooth if z is continuous on [a,b] and if there exist points a = a_{0} < a_{1} < \cdots < a_{n} = b, where z(t) is smooth in the intervals [a_{k}, a_{k+1}] .

The right-hand derivative at a_{k} may differ from the left-hand derivative at a_{k} for k = 1,…,n-1.

(Def) Two parametrizations, z : [a,b] \to \mathbb{C} and \tilde{z} : [c,d] \to \mathbb{C} are equivalent if there exists a continuously differentiable bijection s \mapsto t(s) from [c,d] \to [a,b] so that t’(s) >0 and \tilde{z}(s) = z(t(s)).

(Def) The family of all paramterizations that are equivalent to z(t) determines a smooth curve \gamma \subset \mathbb{C} , namely the image of [a,b] under z with the orientation given by z as t travels from a to b. define a curve \gamma^{-} obtained from the curve \gamma by reversing the orientation.

(Def) It is clear how to define a piecewise-smooth curve. The points z(a) and z(b) are called the end-points of the curve and are independent on the parametrization. Since \gamma carries an orientation, it is natural to say that \gamma begins at z(a) and ends at z(b).

(DeF) A smooth or piecewise smooth curve is closed if z(a) = z(b) for any of its parametrizations. Finaly, a smooth or piecewise-smooth curve is simple if it is not self-intersecting, that is, z(t) \neq z(s) unless s = t.

(Def) For brevity, call any piecewise-smooth curve a curve.

(Def) The positive orientation(counterclockwise) is the one that is given by the standard parametrization z(t) = z_{0} + re^{it} , where t \in [0,2\pi], while the negative orientation(clockwise) is given by z(t) = z_{0} + re^{-it} , where t \in [0,2\pi].

Denote C a general positively oriented circle.

(Def) Given a smooth curve \gamma in \mathbb{C} parametrized by z : [a,b] \to \mathbb{C} , and f a continuous function on \gamma, we define the integral of f along \gamma by \int_{\gamma} f(z) dz = \int_{a}^{b} f(z(t)) z’(t) dt.

(Def) If \gamma is piecewise smooth, if z(t) is a piecewise-smooth parametrization as before, then \int_{\gamma} f(z)dz = \sum_{k = 0}^{n-1} \int_{a_{k}}^{a_{k+1}} f(z(t))z’(t) dt.

(Def) The length of the smooth curve \gamma is length(\gamma) = \int_{a}^{b} |z’(t)| dt.

(Prop 3.1.) Integration of continuous functions over curves satisfies the following properties :

It is linear, that is if \alpha, \beta \in \mathbb{C}, then \int_{\gamma} (\alpha f(z) + \beta g(z)) dz = \alpha \int_{\gamma} f(z)dz + \beta \int_{\gamma} g(z) dz.

If \gamma^{-} is \gamma with the reverse orientation, then \int_{\gamma} f(z) dz = - \int_{\gamma^{-}} f(z) dz.

One has the inequality |\int_{\gamma} f(z) dz | \le \sup_{z \in \gamma} |f(z)} \cdot length(\gamma).

(Def) Suppose f is a function on the open set \Omega. A primitive for f on \Omega is a function F that is holomorphic on \Omega and s.t. F’(z) = f(z) for all z \in \Omega.

(Thm 3.2.) If a continuous function f has a primitive F in \Omega, and \gamma is a curve in \Omega that begins at w_{1} and ends at w_{2}, then \int_{\gamma} f(z) dz = F(w_{2}) – F(w_{1}) .

(Cor 3.3.) If \gamma is a closed curve in an open set \Omega, and f is continuous and has a primitive in \Omega, then \int_{\gamma} f(z)dz = 0.

(Cor 3.4.) IF f is holomorphic in a region \Omega and f’ = 0, then f is constant.

(Def) f(z) = O(g(z)) to mean that there is a constant C > 0 s.t. |f(z)| \le C|g(z)| for z in a neighborhood of the point in question. In addition, we say f(z) = o(g(z)) when |f(z)/g(z) | \to 0. We also write f(z) \simeq g(z) to mean that f(z)/g(z) \to 1.

Cauchy’s Theorem and Its Applications

(Section 2) Cauchy’s Theorem and Its Applications

(subsection 1) Goursat’s Thm

(prop) If f has a primitive in an open set \Omega, then \int_{\gamma} f(z) dz = 0 for any closed curve \gamma in \Omega. Conversely, if we can show that the above relation holds for some types of curves \gamma, then a primitive will exist.

(Thm 11.1.) If \Omega is an open set in \mathbb{C}, and T \subset \Omega a triangle whose interior is also contained in \Omega, then \int_{T} f(z) dz = 0 whenever f is holomorphic in \Omega.

(Cor 1.2.) If f is holomorphic in an open set \Omega that contains a rectangle R and its interior, then \int_{R} f(z) dz = 0.

(subsection 2) Local existence of primitives and Cauchy’s theorem in a disc

(Thm 2.1.) A holomorphic function in an open disc has a primitive in that disc.

(Cauchy’s theorem for a disc) (Thm 2.2.) If f is holomorphic in a disc, then \int_{\gamma} f(z) dz = 0 for any closed curve \gamma in that disc.

(Cor 2.3.) Suppose f is holomorphic in an open set containing the circle C and its interior, Then \int_{C} f(z) dz = 0.

(Def) We call a toy contour any closed curve where the notion of interior is obvious, and a construction similar to that in (Thm 2.1.) is possible in a neighborhood of the curve and its interior. Its positive orientation is that for which the interior is to the left as we travel along the toy contour.

(Def) Keyhole \Gamma . It consists of two almost complete circles one large and one small, connected by a narrow corridor. The interior of \Gamma, which we denote by \Gamma_{int}, is clearly that region enclosed by the curve.

(Prop) For a toy contour \gamma we have that \int_{\gamma} f(z)dz = 0, whenever f is holomorphic in an open set that contains the contour \gamma and its interior.

(subsection 4) Cauchy’s integral formulas

(Thm 4.1.) Suppose f is holomorphic in an open set that contains the closure of a disc D. If C denotes the boundary circle of this disc with the positive orientation, then f(z) = \frac{1}{2\pi i} \int_{C} \frac{f(\zeta)}{\zeta - z} d\zeta for any point z \in D.

(Prop) If f is holomorphic in an open set that contains a (positively oriented) rectangle R and its interior, then f(z) = \frac{1}{2\pi i} \int_{R} \frac{f(\zeta)}{\zeta - z} d\zeta whenever z belongs to the interior of R.

(Cor 4.2.) If f is holomorphic in an open set \Omega, then f has infinitely many complex derivatives in \Omega. Moreover, if C \subset \Omega is a circle whose interior is also contained in \Omega, then f^{(n)} (z) = \frac{n!}{2\pi i} \int_{C} \frac{f(\zeta)}{(\zeta – z)^{n+1}} d \zeta for all z in the interior of C.

(Cauchy inequalities) (Cor 4.3.) If f is holomorphic in an open set that contains the closure of a disc D centered at z_{0} and of radius R, then |f^{(n)} (z_{0})| \le \frac{n!\Vert f \Vert_{C}}{R^{n}} where \Vert f \Vert_{C} = \sup_{z \in C}|f(z)| denotes the supremum of |f| on the boundary circle C.

(Thm 4.4.) Suppose f is holomorphic in an open set \Omega. IF D is a disc centered at z_{0} and whose closure is contained in \Omega, then f has a power series expansion at z_{0} f(z) = \sum_{n = 0}^{\infty} a_{n} (z-z_{0})^{n} for all z \in D, and the coefficients are given by a_{n} = \frac{f^{(n)} (z_{0})}{n!} for all n \ge 0.

(Liouville’s theorem) (Cor 4.5.) If f is entire and bounded, then f is constant.

(Cor 4.6.) Every non-constant polynomial P(z) = a_{n}z^{n} + \cdot + a_{0} with complex coefficients has a root in \mathbb{C}.

(Cor 4.7.) Every polynomial P(z) = a_{n}z^{n} + \cdots + a_{0} of degree n \ge 1 has precisely n roots in \mathbb{C}. If these roots are denoted by w_{1}, …, w_{n}, then P can be factored as P(z) = a_{n}(z-w_{1})(z-w_{2})\cdot(z-w_{n}).

(Thm 4.8.) Suppose f is a holomorphic function in a region \Omega that vanishes on a sequence of distinct points with a limit point in \Omega. Then f is identically 0.

(Cor 4.9) Suppose f and g are holomorphic in a region \Omega and f(z) = g(z) for all z in some non-empty open subset of \Omega (or more generally for z in some sequence of distinct points with limit points in \Omega). Then f(z) = g(z) throughout \Omega.

(subsection 5) Further applications

(subsubsection 5.1.) Morera’s Theorem

(Thm 5.1.) Suppose f is a continuous function in the open disc D s.t. for any triangle T contained in D \int_{T} f(z)dz = 0, then f is holomorphic.

(subsubsection 5.2.) Sequences of holomorphic functions

(Thm 5.2.) If {f_{n}}_{n=1}^{\infty} is a sequence of holomorphic functions that converges uniformly to a function f| in every compact subset of \Omega, then f is holomorphic in \Omega.

(Thm 5.3.) Under the hypotheses of the previous theorem, the sequence of derivatives {f’{n}}{n=1}^{\infty} converges uniformly to f’ on every compact subset of \Omega.

(subsubsection 5.3.) Holomorphic functions defined in terms of integrals

(Thm 5.4.) Let F(z,s) be defined for (z,s) \in \Omega \times [0,1] where \Omega is an open set in \mathbb{C}. Suppose F satisfies the following properties :

F(z,s) is holomorphic in z for each s.

F is continuous on \Omega \times [0,1].

Then the function f defined on \Omega by f(z) = \int_{0}^{1} F(z,s) ds is holomorphic.

(subsubseciton 5.4.) Schwarz reflection principle

(Def) Let \Omega be an open subset of \mathbb{C} that is symmetric with respect to the real line, that is z \in \Omega iff \bar{z} \in \Omega. Let \Omega^{+} denote the part of \Omega that lies in the upper half-plane and \Omega^{-} that part lies in the lower half-plane.

Let I = \Omega \cap \mathbb{R} so that I denotes the interior of that part of the boundary of \Omega^{+} and \Omega^{-} that lies on the real axis. Then we have \Omega^{+} \cap I \cap \Omega^{-} = \Omega .

(Symmetry Principle) (Thm 5.5.) If f^{+} and f^{-} are holomorphic functions in \Omega^{+} and \Omega^{-} respectively that extend continuously to I and f^{+}(x) = f^{-}(x) for all x \in I, then the function f defined on \Omega by f(z) = \begin{cases} f^{+}(z) & if z \in \Omega^{+} \ f^{+}(z) = f^{-}(z) & if z \in I \ f^{-}(z) & if z \in \Omega^{-}\end{cases} is holomorphic on all of \Omega.

(Schwarz reflection principle) (Thm 5.6.) Suppose that f is a holomorphic function in \Omega^{+} that extends continuously to I and s.t. f is real-valued on I. Then there exists a function F holomorphic in all of \Omega s.t. F = f on \Omega^{+}.

(subsubsection 5.5.) Runge’s approximation theorem

(Thm 5.7.) Any function holomorphic in a neighborhood of a compact set K can be approximated uniformly on K by rational functions whose singularities are in K^{c}.

If K^{c} is connected, any function holomorphic in a neighborhood of K can be approximated uniformly on K by polynomials.

(Lem 5.8.) Suppose f is holomorphic in an open set \Omega, and K \subset \Omega is compact. Then, there exists finitely many segments \gamma_{1}, …, \gamma_{N} in \Omega – K s.t. f(z) = \sum_{n = 1}^{N} \frac{1}{2\pi i} \int_{\gamma_{n}} \frac{f(\zeta)}{\zeta - z} d \zeta for all z \in K.

(Lem 5.9.) For any line segment \gamma entirely contained in \Omega – K , there exists a sequence of rational functions with singularities on \gamma that approximate the integral \int_{\gamma}f(\zeta) /(\zeta – z) d\zeta uniformly on K.

(Lem 5.10.) If K^{c} is connected and z_{0} \notin K, then the function 1/(z-z_{0}) can be approximated uniformly on K by polynomials.

Meromorphic Functions and the Logarithm

(Section 3) Meromorphic Functions and the Logarithm

(subsection 1) Zeros and poles

(Def) A point singularity of a function f is a complex number z_{0} s.t. f is defined in a neighborhood of z_{0} but not at the point z_{0} itself. We also call such points isolated singularities.

A complex number z_{0} is a zero for the holomorphic function f if f(z_{0}) = 0. In particular, analytic continuation shows that the zeros of a non-trivial holomorphic functions are isolated.

(prop) If f is holomorphic in \Omega and f(z_{0}) = 0 for some z_{0} \in \Omega, then there exists an open neighborhood U of z_{0} s.t. f(z) \neq 0 for all z \in U – {z_{0}} , unless f is identically zero.

(Thm 1.1.) Suppose that f is holomorphic in a connected open set \Omega, has a zero at a point z_{0} \in Omega, and does not vanish identically in \Omega. Then there exists a neighborhood U \subset \Omega of z_{0} , a non-vanishing holomorphic function g on U, and a unique positive integer n s.t. f(z) = (z-z_{0})^{n}g(z) for all z \in U.

(Def) in the case of above theorem, we say that f has a zero of order n (or multiplicity n) at z_{0}. If a zero is of order 1, we say that it is simple.

(Def) a deleted neighborhood of z_{0} to be an open disc centered at z_{0}, minus the point z_{0}, that is, the set {z: 0 < |z-z_{0}| < r} for some r>0. Then, we say that a function f defined in a deleted neighborhood of z_{0} has a pole at z_{0}, if the function 1/f, defined to be zero at z_{0}, is holomorphic in a full neighborhood of z_{0}.

(Thm 1.2.) If f has a pole at z_{0} \in \Omega, then in a neighborhood of that point there exists a non-vanishing holomorphic function h and a unique positive integer n s.t. f(z) = (z-z_{0})^{-n} h(z).

(Def) the integer n is called the order(or multiplicity) of the pole, and describes the rate at which the function grows near z_{0}. If the pole is of order 1, we say that it is simple.

(Thm 1.3.) If f has a pole of order n at z_{0}, then f(z) = \frac{a_{-n}}{(z-z_{0})^{n}} + \frac{a_{-n+1}}{(z-z_{0})^{n-1}} + \cdots + \frac{a_{-1}}{(z-z_{0})} + G(z), where G is a holomorphic function in a neighborhood of z_{0}.

(Def) The sum \frac{a_{-n}}{(z-z_{0})^{n}} + \frac{a_{-n+1}}{(z-z_{0})^{n-1}} + \cdots + \frac{a_{-1}}{(z-z_{0})} is called the principal part of f at the pole z_{0} , and the coefficient a_{-1} is the residue of f at that pole. We write res_{z_{0}} f = a_{-1}.

(Thm 1.4.) If f has a pole of order n at z_{0}, then res_{z_{0}} f = \lim_{z \to z_{0}} \frac{1}{(n-1)!} (\frac{d}{dz})^{n-1} (z-z_{0})^{n} f(z).

(subsection 2) The residue formula

(Thm 2.1.) Suppose that f is holomorphic in an open set containing a circle C and its interior, except for a pole at z_{0} inside C. Then \int_{C} f(z)dz = 2\pi i res_{z_{0}} f.

(Cor 2.2.) Suppose that f is holomorphic in an open set containing a circle C and its interior, except for poles at the points z_{1}, …, z_{N} inside C. Then \int_{C} f(z) dz = 2\pi i \sum_{k = 1}^{N} res_{z_{k}} f.

(Cor 2.3.) Suppose that f is holomorphic in an open set containing a toy contour \gamma and its interior, except for poles at the points z_{1},, …, z_{N} inside \gamma. Then \int_{\gamma} f(z) dz = 2\pi i \sum_{k = 1}^{N} res_{z_{k}} f.

(Def) The identity \int_{\gamma} f(z) dz = 2\pi i \sum_{k = 1}^{N} res_{z_{k}} f is referred to as the residue formula.

(subsection 3) Singularities and meromorphic functions

(Def) Let f be a function holomorphic in an open set \Omega except possibly at one point z_{0} in \Omega. If we can define f at z_{0} in such a way that f becomes holomorphic in all of \Omega, we say that z_{0} is a removable singularity for f.

(Riemann’s theorem on removable singularities) (Thm 3.1.) Suppose that f is holomorphic in an open set \Omega except possibly at a point z_{0} in \Omega. If f is bounded on \Omega – {z_{0}} , then z_{0} is a removable singularity.

(Cor 3.2.) Suppose that f has an isolated singularity at the point z_{0}. Then z_{0} is a pole of f iff |f(z)| \to \infty as z \to z_{0}.

(Def) Isolated singularities belong to one of three categories : Removable singularities (f bounded near z_{0}), Pole singularities (|f(z)| \to \infty as z \to z_{0}) , Essential singularities. Any singularity that is not removable or a pole is defined to be an essential singularity.

(Casorati-Weierstrass) (Thm 3.3.) Suppose f is holomorphic in the punctured disc D_{r}(z_{0}) – {z_{0}} and has an essential singularity at z_{0}. Then, the image of D_{r}(z_{0}) – {z_{0}} under f is dense in the complex plane.

(Def) A function f on an open set \Omega is meromorphic if there exists a sequence of points {z_{0}, z_{1},z_{2},…} that has no limit poits in \Omega, and such that

the function f is holomorphic in \Omega – {z_{0}, z_{1}, z_{2}, …}, and

f has poles at the points {z_{0}, z_{1}, z_{2},…} .

If f is holomorphic for all large values of z, we consider F(z) = f(1/z), which is now holomorphic in a deleted neighborhood of the origin. We say that f has a pole at infinity if F has a pole at the origin. Similarly, we can speak of f having an essential singularity at infinity, or a removable singularity (hence holomorphic) at infinity in terms of the corresponding behavior of F at 0. A meromorphic function in the complex plane that is either holomorphic at infty or has a pole at infinity is said to be meromorphic in the extended complex plane.

(Thm 3.4.) The meromorphic functions in the extended complex plane are the rational functions.

(Def) Consider the Euclidean space \mathbb{R}^{3} with coordinates (X,Y,Z) where the XY-plane is identified with \mathbb{C} . We denote by \mathbb{S} the sphere centered at (0,0,1/2) and of radius 1/2; Also, we let \mathcal{N} = (0,0,1) denote the north pole of the sphere.

(Def) Given any point W = (X,Y,Z) on \mathbb{S} different from the north pole, the line joining \mathcal{N} and W intersects the XY-plane in a single point which we denote by w = x + iy; w is called the stereographic projection of W. This geometric construction gives a bijective correspondence between points on the punctured sphere \mathbb{S} – {\mathcal{N}} and the complex plane; It is described analytically by the formulas x = \frac{X}{1-Z} and y = \frac{Y}{1-Z} giving w in terms of W, and X = \frac{x}{x^{2} + y^{2} + 1}, Y = \frac{y}{x^{2} + y^{2} + 1} , and Z = \frac{x^2 + y^{2} }{x^{2} + y^{2} + 1} giving W in terms of w.

(Def) Identifying infinity with the point \mathcal{N} on \mathbb{S}, we see that the extended complex plane can be visualized as the full two-dimensional sphere \mathbb{S}; this is the Riemann sphere. Since this construction takes the unbounded set \mathbb{C} into the compact set \mathbb{S} by adding one point, the Riemann sphere is sometimes called the one-point compactification of \mathbb{C}.

(subsection 4) The argument principle and applications

(Argument principle) (Thm 4.1.) Suppose f is meromorphic in an open set containing a circle C and its interior. If f has no poles and never vanishes on C, then \frac{1}{2\pi i} \int_{C} \frac{f’(z)}{f(z)}dz = (number of zeros of f inside C) minus (number of poles of f inside C), where the zeros and poles are counted with their multiplicities.

(Cor 4.2.) The above theorem holds for toy contours.

(Rouche’s theorem) (Thm 4.3.) Suppose that f and g are holomorphic in an open set containing a circle C and its interior. If |f(z)| > |g(z)| for all z \in C, then f and f+g have the same number of zeros inside the circle C.

(Def) A mapping is said to be open if it maps open sets to open sets.

(Open mapping theorem) (Thm 4.4.) If f is holomorphic and non-constant in a region \Omega, then f is open.

(Def) Refer to the maximum of a holomorphic function f in an open set \Omega as the maximum of its absolute value |f| in \Omega.

(Maximum modulus principle) (Thm 4.5.) If f is a non-constant holomorphic function in a region \Omega, then f cannot attain a maximum in \Omega.

(Cor 4.6.) Suppose that \Omega is a region with compact closure \bar{Omega}. If f is holomorphic on \Omega and continuous on \bar{\Omega} then \sup_{z \in \Omega} |f(z)| \le \sup_{z \in \bar{\Omega} - \Omega} |f(z)|.

(subsection 5) Homotopies and simply connected domains

(Def) Let \gamma_{0} and \gamma_{1} be two curves in an open set \Omega with common end-points. So if \gamma_{0}(t) and \gamma_{1}(t) are two parametrizations defined on [a,b], we have \gamma_{0}(a) = \gamma_{1}(a) = \alpha and \gamma_{0}(b) = \gamma_{1}(b) = \beta. These two curves are said to be homotopic in \Omega if for each 0 \le s \le 1 there exists a curve \gamma_{s} \subset \Omega, parametrized by \gamma_{s}(t) defined on [a,b] , s.t. for every s \gamma_{s}(a) = \alpha and \gamma_{s}(b) = \beta, and for all t \in [a,b] \gamma_{s}(t)|{s = 0} = \gamma{0}(t) and \gamma_{s}(t)|{s=1} = \gamma{1}(t). Moreover, \gamma_{s}(t) should be jointly continuous in s \in [0,1] and t \in [a,b].

(Thm 5.1.) If f is holomorphic in \Omega, then \int_{\gamma_{0}} f(z) dz = \int_{\gamma_{1}} f(z) dz whenever the two curves \gamma_{0} and \gamma_{1} are homotopic in \Omega.

(Def) A region \Omega in the complex plane is simply connected if any two pair of curves in \Omega with the same end-points are homotopic.

(Thm 5.2.) Any holomorphic function in a simply connected domain has a primitive.

(Cor 5.3.) If f is holomorphic in the simply connected region \Omega, then \int_{\gamma} f(z)dz = 0 for any closed curve \gamma in \Omega.

(subsection 6) The complex logarithm

(Def) To make sense of the logarithm as a single-valued function, we must restrict the set on which we define it. This is the so-called choice of a branch or sheet of the logarithm.

(Thm 6.1.) Suppose that \Omega is simply connected with 1 \in \Omega, and 0 \notin \Omega. Then in \Omega there is a branch of the logarithm F(z) = log_{\Omega}(z) so that

F is holomorphic in \Omega,

e^{F(z)} = z for all z \in \Omega,

F(r) = log r whenever r is a real number and near 1.

In other words, each branch \log_{\Omega}(z) is an extension of the standard logarithm defined for positive numbers.

(Def) In the slit plane \Omega = \mathbb{C} – {(-\infty, 0]} we have the principle branch of the logarithm log z = log r + i \theta where z = re^{i\theta} with |\theta| < \pi.

(Thm 6.2.) If f is a nowhere vanishing holomorphic function in a simply connected region \Omega, then there exists a holomorphic function g on \Omega s.t. g(z) = e^{g(z)} .

The function g(z) in the theorem can be denoted by log f(z), and determines a branch of that logarithm.

(subsection 7) Fourier series and harmonic functions

(Thm 7.1.) The coefficients of the power series expansion of f are given by a_{n} = \frac{1}{2\pi r^{n}} \int_{0}^{2\pi} f(z_{0} + re^{i\theta}) e^{-in\theta} d\theta for all n \ge 0 and 0 < r < R. Moreover, 0 = \frac{1}{2\pi r^{n}} \int_{0}^{2 \pi} f(z_{0} + re^{i\theta}) e^{-in\theta} d \theta whenever n < 0.

(Mean-value property) (Cor 7.2.) If f is holomorphic in a disc D_{R}(z_{0}), then f(z_{0}) = \frac{1}{2\pi} \int_{0}^{2\pi} f(z_{0} + re^{i\theta}) d\theta , for any 0 < r < R.

(Cor 7.3.) If f is holomorphic in a disc D_{R}(z_{0}) , and u = Re(f), then u(z_{0}) = \frac{1}{2\pi} \int_{0}^{2\pi} u(z_{0} + re^{i\theta}) d\theta, for any 0 < r < R.

The Fourier Transform

(Section 4) The Fourier Transform

(Def) If f is a function on \mathbb{R} that satisfies appropriate regularity and decay conditions, then its Fourier transform is defined by \hat(\xi) = \int_{-\infty}^{\infty} f(x) e^{-2\pi ix \xi} dx , \xi \in \mathbb{R}, and its counterpart, the Fourier inversion formula, holds f(x) = = \int_{-\infty}^{\infty} \hat{f}(\xi) e^{2\pi ix \xi} d\xi

(Def) The condition, given by the Paley-Wiener theorem, is that there be a holomorphic extension of f to \mathbb{C} that satisfies the growth condition |f(z)| \le Ae^{2\piM|z|} for some constant A>0. Functions satisfying this condition are said to be of exponential type.

(Def) A function f is of moderate decrease if f is continuous and there exists A >0 so that |f(x)| \le A/(1+x^{2}) for all x \in \mathbb{R} . A more restrictive condition is that f \in S, the Schwarz space of testing functions, which also implies that \hat{F} belongs to S.

(subsection 1) The class \mathfrak{F}

(Def) For each a >0 we denote by \mathfrak{F}_{a} the class of all functions f that satisfy the following two conditions :

The function f is holomorphic in the horizontal strip S_{a} = {z \in \mathbb{C} : |Im(z)| <a } .

There exists a constant A >0 s.t. |f(x+iy)| \le \frac{A}{1+x^{2}} for all x \in \mathbb{R} and |y|<a.

(Def) Denote by \mathfrak{F} the class of all functions that belong to \mathfrak{F}_{a} for some a.

(subsection 2) Action of the Fourier transform on \mathfrak{F}

(Thm 2.1.) If f belongs to the class \mathfrak{F}_{a} for some a > 0 , then |\hat{f}(\xi)| \le Be^{-2\pib |\xi|} for any 0 \le b < a.

(Thm 2.2.) If f \in \mathfrak{F}, then the Fourier inversion holds, namely f(x) = \int_{-\infty}^{\infty} \hat{f}(\xi) e^{2\pi ix \xi}d \xi for all x \in \mathbb{R}.

(Lem 2.3.) If A is positive and B is real, then \int_{0}^{\infty} e^{-(A+iB)\xi} d\xi = \frac{1}{A+iB}.

(Thm 2.4.) If f \in \mathfrak{F}, then \sum_{n \in \mathbb{Z}}f(n) = \sum_{n \in \mathbb{Z}} \hat{f}(n).

(subsection 3) Paley-Wiener theorem

(Thm 3.1.) Suppose \hat{f} satisfies the decay condition |\hat{f}(\xi)} \le Ae^{-2\pi a |\xi|} for some constants a, A >0. Then f(x) is the restriction to \mathbb{R} of a function f(z) holomorphic in the strip S_{b} = {z \in \mathbb{C} : |Im(z)|<b}, for any 0 < b < a.

(Cor 3.2.) If \hat{f} (\xi) = O(e^{-2\pi a |\xi|} ) for some a >0 , and f vanishes in a non-empty open interval, then f = 0.

(Thm 3.3.) Suppose f is continuous and of moderate decrease on \mathbb{R}. Then f has an extension to the complex plane that is entire with |f(z)| \le Ae^{2\piM|z|} for some A>0, iff \hat{f} is supported in the interval [-M,M].

(Thm 3.4.) Suppose F is a holomorphic function in the sector S = {z : -\pi/4 < arg z < \pi/4 } that is continuous on the closure of S. Assume |F(z)| \le 1 on the boundary of the sector, and that there are constants C, c>0 s.t. |F(z)| \le Ce^{c|z|} for all z in the sector. Then |F(z)| \le 1 for all z \in S.

(Thm 3.5.) Suppose f and \hat{f} have moderate decrease. Then \hat{f} (\xi) = 0 for all \xi < 0 iff f can be extended to a continuous and bounded function in the closed upper half-plane {z = x + iy : y \ge 0} with f holomorphic in the interior.

(section 5) Entire Functions

(subsection 1) Jensen’s formula

(Thm 1.1.) Let \Omega be an open set that contains the closure of a disc D_{R} and suppose that f is holomorphic in \Omega, f(0) \neq 0, and f vanishes nowhere on the circle C_{R}. If z_{1},…,z_{N} denote the zeros of f inside the disc(counted with multiplicities). then log |f(0)| = \sum_{k = 1}^{N} log(\frac{|z_{k}|}{R}) + \frac{1}{2\pi} \int_{0}^{2\pi} log |f(Re^{i\theta})| d\theta.

(Lem 1.2.) If z_{1},…,z_{N} are the zeros of f inside the disc D_{R}, then \int_{0}^{R} n(r) \frac{dr}{r} = \sum_{k=1}^{N} log |\frac{R}{z_{k}}|.

(subsection 2) Functions of finite order

(Def) Let f be an entire function. If there exists a positive number \rho and constants A,B >0 s.t. |f(z)| \le Ae^{B|z|^{\rho}} for all z \in \mathbb{C}, then we say that f has an order of growth \le \rho. We deine the order of growth of f as \pho_{f} = \inf \tho, where the infimum is over all \pho >0 s.t. f has an order of growth \le \rho.

(Thm 2.1.) If f is an entire function that has an order of growth \le \rho, then:

n(r) \le Cr^{\rho} for some C>0 and all sufficiently large r.

If z_{1},z_{2},… denote the zeros of f, with z_{k} \neq 0, then for all s > \rho we have \sum_{k=1}^{\infty} \frac{1}{|z_{k}|^{s}} < \infty.

(subsection 3) Infinite products

(subsubsection 3.1.) Generalities

(Def) Given a sequence {a_{n}}{n=1}^{\infty} of complex numbers, we say that the product \prod{n=1}^{\infty} (1+a_{n}) converges if the limit \lim_{N \to \infty} \prod_{n=1}^{\infty}(1+a_{n}) of the partial products exists.

(Prop 3.1.) If \sum |a_{n}| < \infty, then the product \prod_{n=1}^{\infty} (1+a_{n}) converges. Moreover, the product converges to 0 iff one of its factors is 0.

(prop 3.2.) Suppose {F_{n}} is a sequence of holomorphic functions on the open set \Omega. If there exist constatns c_{n} >0 s.t. \sum c_{n} < \infty and |F_{n}(z) – 1| \le c_{n} for all z \in \Omega, then

The product \prod_{n =1}^{\infty} F_{n}(z) converges unifotmly in \Omega to a holomorphic function F(z).

If F_{n}(z) does not vanish for any n, then \frac{F’(z)}{F(z)} = \sum_{n=1}^{\infty} \frac{F’{n}(z)}{F{n}(z)}.

(subsubseciton 3.2.) Example : The product formula for the sine function

(prop) \frac{sin \pi z}{\pi} = z \prod_{n=1}^{\infty} (1-\frac{z^{2}}{n^{2}}).

(subsection 4) Weierstrass infinite products

(Thm 4.1.) Given any sequence {a_{n}} of complex numbers with |a_{n}| \to \infty as n \to \infty, there exists an entire function f that vanishes at all z = a_{n} and nowhere else. Any other such entire function is of the form f(z) e^{g(z)}} , where g is entire.

(Def) For each integer k \ge 0 we define canonical factors by E_{0}(z) = 1-z and E_{k}(z) = (1-z)e^{z + z^{2}/2 + \cdot + z^{k}/k}, for k \ge 1. The integer k is called the degree of the canonical factor.

(Lem 4.2.) If |z| \le 1/2, then |1-E_{k}(z)| \le c|z|^{k+1} for some c>0.

(subsection 5) Hadamard’s factorization theorem

(Thm 5.1.) Suppose f is entire and has growth order \rho_{0}. Let k be the integer so that k \le \rho_{0} < k+1. If a_{1}, a_{2}, … denote the (non-zero) zeros of f, then f(z) = e^{P(z)} z^{m} \prod_{n=1}^{\infty} E_{k}(z/a_{n}) , where P is a polynomial of degree \le k, and m is the order of the zero of f at z = 0.

(Lem 5.2.) The canonical products satisfy |E_{k}(z)| \ge e^{-c|z|^{k+1}} if |z| \le 1/2 and |E_{k}(z)} \ge |1-z| e^{-c’|z|^{k}} if |z| \ge 1/2.

(Lem 5.3.) For any s with \rho_{0} < s < k+1, we have |\prod_{n=1}^{\infty} E_{k} (z/a_{n})| \ge e^{-c|z|^{s}}, except possibly when z belongs to the union of the discs centered at a_{n} of radius |a_{n}|^{-k-1}, for n = 1,2,3,…

(Cor 5.3.) There exists a sequence of radii, r_{1},r_{2},…, with r_{m} \to \infty, s.t. |\prod_{n=1}^{\infty} E_{k}(z/a_{n})| \ge e^{-c|z|^{s}} for |z| = r_{m}.

(Lem 5.5.) Suppose g is entire and u = Re(g) satisfies u(z) \le Cr^{s} whenever |z| = r for a sequence of positive real numbers r that tends to infinity. Then g is polynomial of degree \le s.

The Gamma and Zeta functions

(Section 6) The Gamma and Zeta functions

(subsection 1) The gamma function

(Def) For s >0, the gamma function is defined by \Gamma(s) = \int_{0}^{\infty} e^{-t}t^{s-1} dt.

(Prop 1.1.) The gamma function extends to an analytic function in the half-plane Re(s) >0, and is still given there by the integral formula as defined.

(subsubsection 1.1.) Analytic continuation

(Lem 1.2.) If Re(s) >0, then \Gamma(s+1) = s \Gamma(s) as a consequence \Gamma(n+1) = n! for n = 0,1,2,…

(Thm 1.3.) The function \Gamma(s) initially defined for Re(s) >0 has an analytic continuation to a meromorphic function on \mathbb{C} whose only singularities are simple poles at the negative integers s = 0, -1, …. The residue of \Gamma at s = -n is (-1)^{n}/n!.

(subsection 1.2.) Further properties of \Gamma

(Thm 1.4.) For all s \in \mathbb{C}, \Gamma(s) \Gamma(1-s) = \frac{\pi}{sin \pi s}.

(Lem 1.5.) For 0<a<1, \int_{0}^{\infty}\frac{v^{a-1}}{1+v} dv = \frac{\pi}{sin \pi a} .

(Thm 1.6.) The function \Gamma has the following properties : 1/\Gamma(s) is an entire function of s with simple zeros at s = 0,-1,-2,… and it vanishes nowhere else.

1/\Gamma(s) has growth |\frac{1}{\gamma(s)}| \le c_{1} e^{c_{2} |s| log|s|}. Therefore, 1/\Gamma is of order 1 in the sense that for every \epsilon >0, there exists a bound c(\epsilon) so that |\frac{1}{\Gamma(s)}| \le c(\epsilon)e^{c_{2}}|s|^{1+\epsilon}.

(Thm 1.7.) For all s \in \mathbb{C}, \frac{1}{\Gamma(s)} = e^{\gamma s} s \prod_{n = 1}^{\infty} (1+\frac{s}{n}) e^{-s/n}. The real number \gamma, which is known as Euler’s constant, is defined by \gamma = \lim_{N \to \infty} \sum_{n = 1}^{N} \frac{1}{n} – log N.

(subsection 2) The zeta function

(Def) The Riemann zeta function is initially defined for real s >1 by the convergent series \zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^{s}}.

(subsubsection 2.1.) Functional equation and analytic continuation

(Prop 2.1.) The series defining \zeta(s) converges for Re(s) >1, and the function \zeta is holomorphic in this half-plane.

(Def) Consider the theta function, which is defined for real t>0 by \vartheta (t) = \sum_{n = -\infty}^{\infty} e^{-\pi n^{2} t}.

(Thm 2.2.) If Re(s) >1, then \pi^{-s/2} \Gamma(s/2) \zeta(s) = \frac{1}{2} \int_{0}^{\infty} u^{(s/2)-1} [\vartheta(u) -1] du.

(Def) The modification of the \zeta function called the xi function, which makes the former appear more symmetric. It is defined for Re(s) >1 by \xi(s) = \pi^{-s/2} \Gamma(s/2) \zeta(s).

(Thm 2.3.) The function \xi is holomorphic for Re(s) >1 and has an analytic continuation to all of \mathbb{C} as a meromorphic function with simple poles at s = 0 and s = 1. Moreover, \xi(s) = \xi (1-s) for all s \in \mathbb{C}.

(Thm 2.4.) The zeta function has a meromorphic continuation into the entire complex plane, whose only singularity is a simple pole at s = 1.

(Prop 2.5.) There are a sequence of entire functions {\delta_{n}(s)}{n=1}^{\infty} that satisfy the estimate |\delta{n}(s)| \ge |s|/n^{\sigma +1} , where s = \sigma + it , and such that \sum_{1 \le n < N} \frac{1}{n^{s}}-\infty_{1}^{N} \frac{dx}{x^{s}} = \sum_{1 \le n < N} \delta_{n}(s), whenever N is an integer >1.

(Cor 2.6.) For Re(s) >0 we have \zeta(s) - \frac{1}{s-1} = H(s) , where H(s) = \sum_{n = 1}^{\infty} \delta_{n}(s) is holomorphic in the half-plane Re(s) >0.

(Prop 2.7.) Suppose s = \sigma + it with \sigma, t \in \mathbb{R}. Then for each \sigma_{0}, 0 \le \sigma_{0} \le 1 , and every \epsilon >0, there exists a constant c_{\epsilon} so that |\zeta(s)| \le c_{\epsilon} |t|^{1-\sigma_{0} + \epsilon}, if \sigma_{0} \le \sigma and |t| \ge 1.

|\zeta’(s)| \le c_{\epsilon} |t|^{\epsilon}, if 1 \le \sigma and |t| \ge 1.

The zeta function and prime number theorem

(section 7) The zeta function and prime number theorem

(subsection 1) Zeros of zeta function

(prop) \zeta(s) = \prod_{p} \frac{1}{1-p^{-s}} .

\zeta(s) = \pi^{s-1/2} \frac{\Gamma((1-s)/2)}{\Gamma(s/2)} \zeta(1-s)

(Thm 1.1.) The only zeros of \zeta outside the strip 0\le Re(s) \le 1 are at the negative even integers, -2,-4,-6,… The region that remains to be studied is called the critical strip.

(Def) Riemann Hypothesis : The zeros of \zeta(s) in the critical strip lie on the line Re(s) = 1/2.

the zeros of \zeta located outside the critical strip are sometimes called the trivial zeros of the zeta function.

(Thm 1.2.) The zeta function has no zeros on the line Re(s) = 1.

(Lem 1.3.) If Re(s) >1, then log \zeta(s) = \sum_{p,m} \frac{p^{-ms}}{m} = \sum_{n=1}^{\infty}c_{n}n^{-s} for some c_{n} \ge 0.

(Cor 1.5.) If \sigma >1 and t is real, then log|\zeta^{3}(\sigma) \zeta^{4}(\sigma + it) \zeta(\sigma + 2it)| \ge 0.

(subsection 1.1.) Estimates for 1/\zeta(s)

(prop 1.6.) For every \epsilon >0, we have 1/|\zeta(s)| \le c_{\epsilon} |t|^{\epsilon} when s = \sigma + it, \sigma \ge 1, and |t| \ge 1.

(section 2) Reduction to the functions \psi and \psi_{1}

(Def) Tchebychev’s \psi-function is defined by \psi(x) = \sum_{p^{m} \le x} log p.

(Prop 2.1.) If \psi(x) \sim x as x \to \infty, then \pi(x) \sim x/log x as x \to \infty.

(Prop 2.2.) If \psi_{1}(x) \sim x^{2}/2 as x \to \infty, then \psi(x) \sim x as x \to \infty, and therefore \pi(x) \sim x/log x as x \to \infty.

(Prop 2.3.) For all c > 1 \psi_{1} (x) = \frac{1}{2\pi i} \int_{c-i\infty}^{c + i\infty} \frac{x^{s+1}}{s(s+1)} (-\frac{\zeta’(s)}{\zeta(s)}) ds.

(Lem 2.4.) If c >0, then \frac{1}{2\pi i} \int_{c-i\infty}^{c + i\infty}\frac{a^{s}}{s(s+1)} ds = \begin{cases}0 & if 0 < a \le 1 \ 1-1/a & if 1\le a. \end{cases}

Here, the integral is over the vertical line Re(s) = c.

(subsubsection 2.1.) Proof of the asymptotic for \psi_{1}

(prop) \psi_{1}(x) \sim x^{2}/2 as x \infty.

(subsubsection) Note on interchanging double sums. if {a_{kl}{1\le k,l < \infty} is a sequence of complex numbers indexed by \mathbb{N} \times \mathbb{N}, s.t. \sum{k=1}^{\infty} (\sum_{l=1}^{\infty} |a_{kl}|) <\infty, then

The double sum A = \sum_{k=1}^{\infty} (\sum_{l=1}^{\infty} a_{kl} ) summed in this order converges, and we may in fact interchange the order of summation, so that A = \sum_{k=1}^{\infty} \sum_{l=1}^{\infty} a_{kl} = \sum_{l=1}^{\infty} \sum_{k=1}^{\infty} a_{kl} .

Given \epsilon >0, there is a positive integer N so that for all K,L >N we have |A-\sum_{k=1}^{K} \sum_{l=1}^{L} a_{kl} | <\epsilon.

If m \mapsto (k(m), l(m)) is a bijection from \mathbb{N} to \mathbb{N} \times \mathbb{N}, and if we write c_{m} = a_{k(m)l(m)}, then A = \sum_{k=1}^{\infty} c_{k}.

(Def) A bijective holomorphic function f : U \to V is called a conformal map or biholomorphism. Given such a mapping f, we say that U and V are conformally equivalent or simply biholomorphic.

(Prop 1.1.) If f : U \to V is holomorphic and injective, then f’(z) \neq 0 for all z \in U. In particular, the inverse of f defined on its range is holomorphic, and thus the inverse of a conformal map is also holomorphic.

(subsubseciton 1.1.) The disc and upper half-plane

(Def) The upper half-plane, which we denote by \mathbb{H}, consists of those complex numbers with positive imaginary part; that is, \mathbb{H} = {z \in \mathbb{C} : Im(z) >0}.

(Thm 1.2.) The map F : \mathbb{H} \to \mathbb{D} : F(z) = \frac{i-z}{i+z} is a conformal map with inverse G : \mathbb{D} \to \mathbb{H} : G(w) = i\frac{1-w}{1+w}.

(Def) Mappings of the form z \mapsto \frac{az+b}{cz+d} where a,b,c, and d are complex numbers, and where the denominator is assumed not to be multiple of the numerator, are usualy referred to as fractional linear transformations.

(subsubsection 1.2.) Further examples

(Def) For any non-zero complex number c, the map f : z \mapsto cz is a conformal map from the complex plane to itself, whose inverse is simply g: w\mapsto c^{-1}w . If c has modulus 1, so that c = e^{i\varphi} for some real \varphi, then f is a rotation by \varphi. If c >0, then f corresponds to a dilation.

(subsubsection 1.3.) The Dirichlet problem in a strip

(Def) The Dirichlet problem in the open set \Omega consists of solving \begin{cases} \Delta u = 0 & in \Omega \ u = f & on \partial \Omega, \end{cases} where \Delta denotes the Laplacial \partial^{2}/\partial x^{2} + \partial^{2}/\partial y^{2} , and f is a given function on the boundary of \Omega .

(Lem 1.3.) Let V and U be open sets in \mathbb{C} and F : V \to U a holomorphic function. If u : U \to \mathbb{C} is a harmonic function, then u \bullet F is harmonic on V.

(subsection 2) The Schwarz lemma ; automorphisms of the disc and upper half-plane

(Lem 2.1.) Let f : \mathbb{D} \to \mathbb{D} be holomorphic with f(0) = 0. Then |f(z)| \le |z| for all z \in \mathbb{D}.

If for some z_{0} \neq 0 we have |f(z_{0})| = |z_{0}| , then f is a rotarion.

|f’(0)| \le 1, and if equality holds, then f is a rotation.

(subsubsection 2.1.) Automorphisms of the disc

(Def) A conformal map from an open set \Omega to itself is called an automorphism of \Omega. The set of all automorphisms of \Omega is denoted by Aut(\Omega), and carries the structure of a group.

if f and g are automorphisms of \Omega, then f \bullet g is also an automorphism.

(Thm 2.2.) If f is an automorphism of the disc, then there exist \theta \in \mathbb{R} and \alpha \in \mathbb{D} s.t. f(z) = e^{i\theta} \frac{\alpha - z}{1-\bar{\alpha} z}.

(Cor 2.3.) The only automorphisms of the unit disc that fix the origin are the rotations.

(subsubsection 2.2.) Automorphisms of the upper half-plane

(prop ) Aut(\mathbb{D}) and Aut(\mathbb{H}) are the same.

Aut(\mathbb{H}) consists of all maps z \mapsto \frac{az + b}{cz + d} where a,b,c and d are real numbers with ad – bc = 1.

(Def) Let SL_{2} (\mathbb{R}) denote the group of all 2 \times 2 matrices with real entreis and determinant 1, namely SL_{2}(\mathb{R}) = {M = \begin{pmatrix}a & b \ c & d \end{pmatrix} : a,b,c,d \in \mathbb{R} and det(M) = ad-bc = 1}. This group is called the special linear group.

Given a matrix M \in SL_{2}(\mathbb{R}) we define the mapping f_{M} by f_{M}(z) = \frac{az+b}{cz+d}.

(Thm 2.4.) Every automorphism of \mathbb{H} takes the form f_{M} for some M \in SL_{2}(\mathbb{R}) . Conversely, every map of this form is an automorphism of \mathbb{H}.

(Def) Since the two matrices M and -M give rise to the same function f_{M} = f_{-M}, if we identify the two matrices M and -M, then we obtain a new group PSL_{2}(\mathbb{R}) called the projective special linear group; which is isomorphic with Aut(\mathbb{H}) .

(subsection 3) The Riemann mapping theorem

(subsubsection 3.1.) Necessary conditions and statement of the theorem

(Def) Call a subset \Omega of \mathbb{C} proper if it is non-empty and not the whole of \mathbb{C} .

(Riemann mapping theorem) (Thm 3.1.) Suppose \Omega is proper and simply connected. If z_{0} \in \Omega, then there exists a unique conformal map F : \Omega \to \mathbb{D} s.t. F(z_{0}) = 0 and F’(z_{0}) >0.

(Cor 3.2.) Any two proper simply connected open subsets in \mathbb{C} are comformally equivalent.

(subsubsection 3.2.) Montel’s theorem

(Def) Let \Omega be an open subset of \mathbb{C}. A family \mathcal{F} of holomorphic functions on \Omega is said to be normal if every sequence in \mathcal{F} has a subsequence that converges uniformly on every compact subset of \Omega.

The family \mathcal{F} is said to be uniformly bounded on compact subsets of \Omega if for each compact set K \subset \Omega there exists B >0 s.t. |f(z)| \le B for all z \in K and f \in \mathcal{F}.

The family \mathcal{F} is equicontinuous on a compact set K if for every \epsilon >0 there exists \delta >0 s.t. whenever z , w \in K and |z-w| < \delta then |f(z)_f(w)| < \epsilon for all f \in \mathcal{F}.

(Thm 3.3.) Suppose \mathcal{F} is a family of holomorphic functions on \Omega that is uniformly bounded on compact subsets of \Omega. Then

\mathcal{F} is equicontinuous on every compact subset of \Omega.

\mathcal{F} is a normal family.

(Def) A sequence {K_{l}}_{l=1}^{\infty} of compact subsets of \Omega is called an exhaustion if

K_{l} is contained in the interor of K_{l+1} for all l = 1,2,…

Any compact set K \subset \Omega is contained in K_{l} for some l. In particular \Omega = \bigcup_{l=1}^{\infty} K_{l}.

(Lem 3.4.) Any open set \Omega in the complex plane has an exhaustion.

(Prop 3.5.) If \Omega is a connected open subset of \mathbb{C} and {f_{n}} a sequence of injective holomorphic functions on \Omega that converges uniformly on every compact subset of \Omega to a holomorphic function f, then f is either injective or constant.

(subsubsection 3.3.) Proof of Riemann mapping theorem

(Def) \Omega is holomorphically simply connected if for any holomorphic function f in \Omega and any closed curve \gamma in \Omega, we have \int_{\gamma} f(z) dz = 0.

(subsection 4) Conformal mappings onto polygons

(subsubsection 4.2.) The Schwarz-Christoffel integral

(Def) The general Schwarz-Christoffel integral by S(z) = \int_{0}^{z} \frac{d\zeta}{(\zeta – A_{1})^{\beta_{1}}\cdot(\zeta – A_{n})^{\beta_{n}}}. Here A_{1} < A_{2} < \cdot < A_{n} are n distinct points on the real axis arranges in increasing order. The exponents \beta_{k} will be assumed to satisfy the conditions \beta_{k} <1 for each k and 1 < \sum_{k=1}^{n} \beta_{k}.

The integrand is defined as follows : (z-A_{k})^{\beta_{k}} is that branch (Defined in the complex plane slit along the infinite ray {A_{k} + iy : y \le 0}) which is positive when z = x is real and x > Z_{k} . As a result (z-A_{k})^{\beta_{k}} = \begin{cases} (x-A_{k})^{\beta_{k}} & if x is real and x > A_{k} \ |x – A_{k}|^{\beta_{k}} e^{i\pi \beta_{k}} & if x is real and x < A_{k}.\end{cases}

(Prop 4.1.) Suppose S(z) is given as above.

If \sum_{k=1}^{n} \beta_{k} = 2, and \mathfrak{p} denotes the polygon whose vertices are given (in order) by a_{1}, …, a_{n}, then S maps the real axis onto \mathfrak{p} – {a_{\infty}}. The point a_{\infty} lies on the segment [a_{n}, a_{1}] and is the image of the point at infinity. Moreover, the (interior) angle at the vertex a_{k} is \alpha_{k} \pi where \alpha_{k} = 1- \beta_{k}.

There is a similar conclusion when 1< \sum_{k=1}^{n} \beta_{k} <2 except now the image of the extended line is the polygon of n+1 sides with vertices a_{1}, a_{2},… , a_{n} , a_{\infty}. The angle at the vertex a_{\infty} is \alpha_{\infty} \pi with \alpha_{\infty} = 1- \beta_{\infty}, where \beta_{\infty} = 2 - \sum_{k=1}^{n} \beta_{k}

(subsubsection 4.3.) Boundary behavior

(Def) a polygonal region P, namely a bounded, simply connected open set whose boundary is a polygonal line \mathfrak{p}. In this context, we always assume that the polygonal line is closed and we sometimes refer to \mathfrak{p} as a polygon.

(Thm 4.2.) If F : \mathbb{D} \to P is a conformal map, then F extends to a continuous bijection from the closure \bar{\mathbb{D}} of the disc to the closure \bar{P} of the polygonal region. In particular, F gives rise to a bijection from the boundary of the disc to the boundary polygon \mathfrak{p}.

(Lem 4.3.) For each 0 < r < 1/2, let C_{r} denote the circle centered at z_{0} of radius r. Suppose that for all sufficienctly small r we are given two points z_{r} and z’{r} in the unit disc that also lie on C{r}. If we let \rho(r) = |f(z_{r}) – f(z’{r})| , then there exists a sequence {r{n}} of radii that tends to zero, and s.t. \lim_{n \to \infty} \rho(r_{n}) = 0.

(Lem 4.4.) Let z_{0} be a point on the unit circle. Then F(z) tends to a limit as z approaches z_{0} within the unit disc.

(Lem 4.5.) The conformal map F extends to a continuous function from the closure of the disc to the closure of the polygon.

(subsubseciton 4.4.) The mapping formula

(Def) Suppose P is a polygonal region bounded by a polygon \mathfrak{p} whose vertices are ordered consecutively a_{1},a_{2},…,a_{n} and with n \ge 3. We denote by \pi \alpha_{k} the interior angle at a_{k}, and define the exterior angle \pi \beta_{k} by \alpha_{k} + \beta_{k} = 1. A simple geometric argument provides \sum_{k=1}^{n} \beta_{k} = 2.

(Thm 4.6.) There exist complex numbers c_{1} and c_{2} so that the conformal map F of \mathbb{H} to P is given by F(z) = c_{1}S(z) + c_{2} where S is the Schwarz-Christoffel integral introduced before.

(Thm 4.7.) If F is a conformal map from the upper half-plane to the polygonal region P and maps the points A_{1},…,A_{n-1}, \infty to the vertices of \mathfrak{p}, then there exists constants C_{1} and C_{2} s.t. F(z) = C_{1} \int_{0}^{z} \frac{d\zeta}{(\zeta – A_{1})^{\beta_{1}}\cdot(\zeta – A_{n-1})^{\beta_{n-1}}} + C_{2}.

(subsubsection 4.5.) Return to elliptic integrals

(Def) The elliptic integral I(z) = \int_{0}^{z} \frac{d \zeta}{[(1-\zeta^{2})(1-k^{2}\zeta^{2}]^{1/2}} with 0 < k < 1.

(prop)it mapped the real axis to the rectangle R with vertices -K, K, K + i K’, and -K + iK’.

This mapping is a conformal mapping of \mathbb{H} to the interior of R.

An introduction to Elliptic Functions

(section 9) An introduction to Elliptic Functions

(subsection 1) Elliptic functions

(Def) meromorphic functions f on \mathbb{C} that have two periods; there are two non-zaro complex numbers \omega_{1} and \omega_{2} s.t. f(z + w_{1}) = f(z) and f(z+w_{2}) = f(z) for all z \in \mathbb{C} . A function with two periods is said to be doubly periodic.

Assume f is a meromorphic function on \mathbb{C} with periods 1 and \tau where Im(\tau) >0. Successive applications of the periodicity conditions yield f(z + n + m \tau) = f(z) for all integers n,m and all z \in \mathbb{C}, and it is therefore natural to consider the lattice in \mathbb{C} defined by \Lambda = {n + m\tau : n , m \in \mathbb{Z}} . We say that 1 and \tau generate \Lambda.

Associated to the lattice \Lambda is the fundamental parallelogram defined by P_{0} = {z \in \mathbb{C} : z = a + b \tau where 0 \le a < 1 and 0 \le b < 1}.

(Def) Two complex numbers z and w are congruent modulo \Lambda if z = w + n + m \tau for some n , m \in \mathbb{Z}, and we write z \sim w. in other words, z and w differ by a point in the lattice, z – w \in \Lambda.

(Def) A period parallelogram P is any translate of the fundamental parallelogram , P = P_{0} + h with h \in \mathbb{C}.

(Prop) \Lambda and P_{0} give rise to a covering (or tiling) of the complex plane \mathbb{C} = \bigcup_{n , m \in \mathbb{Z}} (n + m\tau + P_{0}), and this union is disjoint.

(Prop 1.1.) Suppose f is a meromorphic function with two periods 1 and \tau which generate the lattice \Lambda. Then :

Every point in \mathbb{C} is congruent to a unique point in the fundamental parallelogram.

Every point in \mathbb{C} is congruent to a unique point in the fundamental parallelogram.

The lattice \Lambda provides a disjoint covering of the complex plane, in the sense of above (prop).

The function f is completely determined by its values in any period parallelograms.

(subsubsection 1.1.) Liouville’s theorems

(Thm 1.2.) An entire doubly periodic function is constant.

(Def) A non-constant doubly periodic meromorphic function is called an elliptic function.

(Thm 1.3.) The total number of poles of an elliptic function in P_{0} is always \ge 2.

(Thm 1.4.) Every elliptic function of order m has m zeros in P_{0}.

(subsubsection 1.2.) The Weierstrass \varphi function

(Def) Let \Lambda^{} denote the lattice minus the origin, that is, \Lambda^{} = \Lambda – {(0,0)}, and consider instead the following series : \frac{1}{z^{2}} + \sum_{\omega \in \Lambda^{*}} [\frac{1}{(z+\omega)^{2}} - \frac{1}{\omega^{2}}].

(Lem 1.5.) The two series \sum_{(n,m) \neq (0,0)} \frac{1}{(|n| + |m|)^{r}} and \sum_{n+m \tau \in \Lambda^{*}} \frac{1}{|n+m\tau|^{r}} converge if r > 2.

(Def) Define Weierstrass \varphi function, which is given by the series \frac{1}{z^{2}} + \sum_{\omega \in \Lambda^{*}} [\frac{1}{(z+\omega)^{2}} - \frac{1}{\omega^{2}}]. = \frac{1}{z^{2}} + \sum_{(n,m) \neq (0,0)}[\frac{1}{(z + n + m \tau}^{2} - \frac{1}{(n+m\tau)^{2}}].

\varphi is a meromorphic function with double poles at the lattice points.

(Thm 1.6.) The function \varphi is an elliptic function that has periods 1 and \tau, and double poles at the lattice points.

(Def) Since \varphi’ is elliptic and has order 3, the three points 1/2, \tau/2, and (1+\tau)/2 (which are called the half-periods) are the only roots of \varphi’ in the fundamental parallelogram, and they have multiplicity 1. Therefore, if we define \varphi(1/2) = e_{1}, \varphi(\tau/2) = e_{2} and \varphi(\frac{1+\tau}{2}) = e_{3}

(Thm 1.7.) The function (\varphi’)^{2} is the cubic polynomial in \varphi (\varphi’)^{2} = 4(\varphi – e_{1}) (\varphi -e_{2}) (\varphi – e_{3}) .

(Thm 1.8.) Every elliptic function f with periods 1 and \tau is a rational function of \varphi and \varphi ‘.

(Lem 1.9.) Every even elliptic function F with periods 1 and \tau is a rational function of \varphi.

(subsection 2) The modular character of elliptic functions and Eisenstein series

(Def) Consider the group of transformations of the upper half-plane Im(\tau) >0, generated by the two transformations \tau \mapsto \tau +1 and \tau \mapsto -1/\tau. This group is called the modular group.

(subsubsection 2.1.) Eisenstein series

(Def) The Eisenstein series of order k is defined by E_{k}(\tau) = \sum_{(n,m) \neq (0,0)} \frac{1}{(n+m\tau)^{k}}, whenever k is an integer \ge 3 and \tau is a complex number with Im(\tau) >0. If \Lambda is the lattice generated by 1 and \tau, and if we write \omega = n + m \tau, then another expression for the Eisenstein series is \sum_{\omega \in \Lambda^{*}} 1/\omega^{k}.

(Thm 2.1.) Eisenstein series have the following properties :

The series E_{k}(\tau) converges if k \ge 3, and is holomorphic in the upper half-plane.

E_{k}(\tau) = 0 if k is odd.

E_{k}(\tau) satisfies the following transformation relations:

E_{k}(\tau +1) = E_{k}(\tau) and E_{k}(\tau) = \tau^{-k} E_{k}(-1/\tau).

The last property is sometimes referred to as the modular character of the Eisenstein series.

(Thm 2.2.) For z near 0, we have \varphi(z) = \frac{1}{z^{2}} + 3E_{4}z^{2} + 5E_{6}z^{4} + \cdot = \frac{1}{z^{2}} + \sum_{k=1}^{\infty} (2k+1) E_{2k+2} z^{2k}.

(Cor 2.3.) If g_{2} = 60E_{4} and g_{3} = 140E_{6}, then (\varphi ‘)^{2} = 4 \varphi^{3} – g_{2} \varphi – g_{3}.

(subsubsection 2.2.) Eisenstein series and divisor functions

(Lem 2.4.) If k \ge 2 and Im(\tau) >0, then \sum_{n=-\infty}^{\infty} \frac{1}{(n+\tau)^{k}} = \frac{(-2\pi i)^{k}}{(k-1)!} \sum_{l=1}^{\infty} l^{k-1} e^{2\pi i \tau l}.

(Def) The divisor function \sigma_{l}(r) that arises here is defines as the sum of the l^{th} powers of the divisors of r, that is , \sigma_{l}(r) = \sum_{d|r} d^{l} .

(Thm 2.5.) If k \ge 4 is even, and Im(\tau) >0, then E_{k}(\tau) = 2 \zeta(k) + \frac{2(-1)^{k/2} (2\pi)^{k}}{(k-1)!}\sum_{r=1}^{\infty} \sigma_{k-1} (r) e^{2\pi i \tau r}.

(Cor 2.6.) The double sum defining F converges in the indicated order. We have F(\tau) = 2 \zeta(2) – 8 \pi^{2} \sum_{r=1}^{\infty} \sigma(r) e^{2\pi i r \tau}, where \sigma(r) = \sum_{d|r} d is the sum of the divisors of r.

Applications of Theta functions

(section 10) Applications of Theta functions

(subsection 1) Product formula for the Jacobi theta function

(Def) Jacobi’s theta function is defined for z \in \mathbb{C} and \tau \in \mathbb{H} by \Theta (z|\tau) = \sum_{n = - \infty}^{\infty} = \sum_{n = - \infty}^{\infty} e^{\pi i n^{2} \tau} e^{2 \pi i n z}.

Two significant special cases are \theta(\tau) and \vartheta(t), which are defined by \theta(\tau) = \sum_{n = -\infty}^{\infty} e^{\pi i n^{2} \tau} , \tau \in \mathbb{H}, \vartheta(t) = \sum_{n = -\infty}^{\infty} e^{-\pi n^{2} t}, t>0.

For example, heat kernel was given by H_{t}(x) = \sum_{n = - \infty}^{\infty} = \sum_{n = - \infty}^{\infty} e^{-4\pi^{2} n^{2} t} e^{2 \pi i n x}, and therefore H_{t}(x) = \Theta(x|4\pi i t).

(Prop 1.1.) The function \Theta satisfies the following properties :

\Theta is entire in z \in \mathbb{C} and holomorphic in \tau \in \mathbb{H}.

\Theta(z+1|\tau) = \Theta(z | \tau).

\Theta(z + \tau | \tau) = \Theta(z|\tau) e^{-\pi i \tau} e^{-2\pi i z}

\Theta(z|\tau) = 0 whenever z = 1/2 + \tau/2 + n + m\tau and n,m \in \mathbb{Z}.

(Def) A product \prod(z|\tau) that enjoys the same structural properties as \Theta(z|\tau) as a function of z. This product is defined for z \in \mathbb{C} and \tau \in \mathbb{H} by \prod(z|\tau) = \prod_{n=1}^{\infty} (1-q^{2n}) (1+ q^{2n-1}e^{2\pi i z}) (1+ q^{2n-1}e^{-2\pi i z}) , where we have used the notation that is standard in the subject, namely q = e^{\pi i \tau}. The function \prod(z|\tau) is sometimes referred to as the triple-product.

(Prop 1.2.) The function \prod(z|\tau) satisfies the following properties :

\prod(z,\tau) is entire in z \in \mathbb{C} and holomorphic for \tau \in \mathbb{H}.

\prod(z+1|\tau) = \prod(z | \tau).

\prod(z + \tau|\tau) = \prod(z|\tau) e^{-\pi i \tau} e^{-2\pi i z}.

\prod(z|\tau) = 0 whenever z = 1/2 + \tau/2 + n + m\tau and n , m \in \mathbb{Z}. Moreover, these points are simple zeros of \prod(\cdot | \tau) , and \prod(\cdot | \tau) has no other zeros.

(Product formula) (Thm 1.3.) For all z \in \mathbb{C} and \tau \in \mathbb{H} we have the identity \Theta(z|\tau) = \prod(z | \tau).

(Cor 1.4.) If Im(\tau)>0 and q = e^{\pi i \tau}, then \theta(\tau) = \prod_{n=1}^{\infty} (1-q^{2n}) (1+ q^{2n-1})^{2}. Thus \theta(\tau) \neq 0 for \tau \in \mathbb{H}.

(Cor 1.5.) For each fixed \tau \in \mathbb{H}, the quotient (log \Theta(z|\tau))” = \frac{\Theta(z|\tau) \Theta”(z|\tau) – (\Theta’(z|\tau))^{2}}{\Theta(z|\tau)^{2}} is an elliptic function of order 2 with periods 1 and \tau, and with a double pole at z = 1/2 + \tau/2.

In the above, the primes ‘ denote differentiation with respect to the z variable.

(subsubsection 1.1.) Further transformation laws

(Thm 1.6.) If \tau \in \mathbb{H}, then \Theta(z|-1/\tau) = \sqrt{\frac{\tau}{i}} e^{\pi i \tau z^{2}} \Theta(z\tau | \tau) for all z \in \mathbb{C}. Here \sqrt{\frac{\tau}{i}} denotes the branch of the square root defined on the upper half-plane, that is positive when \tau = it, t >0.

(Cor 1.7.) If Im(\tau) >0, then \theta(-1/\tau) = \sqrt{\frac{\tau}{i}}\theta(\tau).

(Cor 1.8.) If \tau \in \mathbb{H}, then \theta(1-1/\tau) = \sqrt{\frac{\tau}{i}} \sum_{n=-\infty}^{\infty} e^{\pi i (n+1/2)^{2} \tau} = \sqrt{\frac{\tau}{i}}( 2 e^{\pi i \tau/4} + \cdots). The second identity means that \theta(1-1/\tau) \sim \sqrt{\frac{\tau}{i}} 2e^{i\pi \tau/4} as Im(\tau) \to \infty.

(Def) Dedekind eta function, which is defined for Im(\tau) >0 by \eta(\tau) = e^{\frac{\pi i \tau}{12}} \prod_{n=1}^{\infty} (1- e^{2\pi i n \tau}).

(prop 1.9.) If Im(\tau) >0, then \eta(-1/\tau) = \sqrt{\tau/i} \eta(\tau).

(subsection 2) Generating functions

(Def) Given a sequence {F_{n}}{n=0}^{\infty}, its generating function F(x) = \sum{n=0}^{\infty} F_{n}x^{n}.

(Def) The partition function is defined as follows : if n is a positive integer, we let p(n) denote the numbers of ways n can be written as a sum of positive integers. For instance, p(1) = 1, and p(2) = 2 since 2 = 2+0 = 1+1. We set p(0) = 1.

(Thm 2.1.) If |x| < 1, then \sum_{n=1}^{\infty} p(n) x^{n} = \prod_{k=1}^{\infty} \frac{1}{1-x^{k}}.

(Def) Let p_{e,u}(n) denote the number of partitions of n into an even number of unequal parts, and p_{0,u} (n ) the number of partitions of n into an odd number of unequal parts. Then , Euler proved that, unless n is a pentagonal number, one has p_{e,u}(n) = p_{o,u}(n) . By definition, pentagonal numbers are integers n of the form k(3k+1)/2, with k \in \mathbb{Z}.

(Prop 2.2.) \prod_{n=1}^{\infty}(1-x^{n}) = \sum_{k=-\infty}^{\infty} (-1)^{k}x^{\frac{k(3k+1)}{2}}

(subsection 3) The theorems about sum of squares

(subsubseciton 3.1.) The two-squares theorem

(Def) divisor functions : d_{1}(n) denote the number of divisors of n of the form 4k +1, and d_{3} (n) the number of divisors of n of the form 4k +3.

define r_{2}(n) to be the number of ways n can be written as the sum of two squares, counting obvious repetitions; that is, r_{2}(n) is the number of pairs (x,y) x, y \in \mathbb{Z}, so that n = x^{2} + y^{2} .

Define r_{4}(n) to be the number of ways of expressing n as a sum of four squares.

(Thm 3.1.) If n \ge 1, then r_{2}(n) = 4(d_{1}(n) – d_{3}(n)).

If n = p_{1}^{\alpha_{1}}\cdots p_{r}^{\alpha_{r}} is the prime factorization of n where p_{1},…,p_{r} are distinct, then : The positive integer n can be represented as the sum of two squares iff every prime p_{j} of the form 4k + 3 that occurs in the factorization of n has an even exponent a_{j}.

(Prop 3.2.) The identity r_{2}(n) = 4(d_{1}(n) – d_{3}(n)) , n \ge 1, is equivalent to the identities \theta(\tau)^{2} = 2 \sum_{n=-\infty}^{\infty}\frac{1}{q^{n} + q^{-n}} = 1 + 4 \sum_{n=1}^{\infty} \frac{q^{n}}{1+q^{2n}}. whenever q = e^{\pi i \tau} and \tau \in \mathbb{H}.

(Prop 3.3.) The function \mathcal{C}(\tau) = \sum 1/cos(\pi n \tau) , defined in the upper half-plane, satisfies

\mathcal{C}(\tau +2) = \mathcal{C}(\tau).

\mathcal{C}(\tau) = (i/\tau)\mathcal{C}(-1/\tau).

\mathcal{C}(\tau) \to 1 as Im(\tau) \to \infty.

\mathcal{C}(1-1/\tau) \sim 4(\tau/i) e^{\pi i \tau/2} as Im(\tau) \to \infty.

Moreover, \theta(\tau)^{2} satisfies the same properties.

(Thm 3.4.) Suppose f is a holomorphic function in the upper half-plane that satisfies:

f(\tau +2) = f(\tau),

f(-1/\tau) = f(\tau),

f(\tau) is bounded,

then f is constant.

(Def) A subset of the closed upper half-plane, which is defined by \mathcal{F} = {\tau \in \bar{\mathbb{H}} : |Re(\tau)|\le 1 and |\tau| \ge 1}. The points corresponding to \tau = \mp 1 are called cusps.

(Lem 3.5.) Every point in the upper half-plane can be mapped into \mathcal{F} using repeatedly one or another of the following fractional linear transformations or their inverses: T_{2} : \tau \mapsto \tau +2, S : \tau \mapsto -1/\tau.

For this reason, \mathcal{F} is called the fundamental domain for the group of transformations generated by T_{2} and S.

(Def) We let G denote the group generated by T_{2} and S. Since T_{2} and S are fractional linear transformations, we may represent an element g \in G by a matrix g = \begin{pmatrix} a & b \ c & d \end{pmatrix} with the understanding that g(\tau) = \frac{a\tau + b}{c\tau + d}. Since the matrices representing T_{2} and S have integer coefficients and determinant 1, the same is true for all matrices of elements in G. In particular, if \tau \in \mathbb{H}, then Im(g(\tau)) = \frac{Im(\tau)}{|c\tau + d|^{2}}.

(subsection 3.2.) The four-squares theorem

(Def) \sigma_{1}^{*}(n) equals the sum of divisors of n that are not divisible by 4.

(Thm 3.6.) Every positive integer is the sum of four squares, and moreover r_{4}(n) = 8 \sigma_{1}^{*}(n) for all n \ge 1.

(Prop 3.7.) The assertion r_{4}(n) = 8\signa_{1}^{} (n) is equivalent to the identity \theta(\tau)^{4} = \frac{-1}{\pi^{2}} E_{2}^{}(\tau), where \tau \in \mathbb{H}.

(Prop 3.8.) The function E_{2}^{*}(\tau) defined in the upper half-plane has the following properties :

E_{2}^{}(\tau + 2) = E_{2}^{}(\tau).

E_{2}^{}(\tau) = -\tau^{-2} E_{2}^{{}(-1/\tau).

E_{2}^{*}(\tau) \to -\pi^{2} as Im(\tau) \to \infty.

|E_{2}^{*} (1-1/\tau)| = O(|\tau^{2} e^{\pi i \tau}|) as Im(\tau) \to \infty.

moreover -\pi^{2}\theta^{4} has the same properties.

(Def) The forbidden Eisenstein series F and its reverse \tilde{F}, which is obtained from reversing the order of summation : F(\tau) = \sum_{m} \sum_{n} \frac{1}{(m\tau + n)^{2}} and \tilde{F} (\tau) = \sum_{n} \sum_{m} \frac{1}{(m\tau + n)^{2}}. In both cases, the term n = m = 0 is omitted.

(Lem 3.9.) The functions F and \tilde{F} satisfy :

F(-1/\tau) = \tau^{2} \tilde{F}(\tau),

F(\tau) - \tilde{F}(\tau) = 2\pi i/\tau.

F(-1/\tau) = \tau^{2}F(\tau) – 2\pi i \tau.

Appendix

(Appendix A) Asymptotics

(subsection 1) Bessel functions

(Def) For all order \nu > -1/2, J_{\nu}(s) = \frac{(s/2)^{\nu}}{\Gamma(\nu + 1/2) \Gamma(1/2)} \int_{-1}^{1} e^{isx} (1-x^{2})^{\nu – 1/2} dx.

If we also write J_{-1/2}(s) for \lim_{v\to-1/2} J_{\nu}(s) , it equals \sqrt{\frac{2}{\pi s}} cos s; . J_{1/2}(s) = \sqrt{\frac{2}{\pi s}} sin s.

(Thm 1.1.) J_{\nu}(s) = \sqrt{2}{\pi s} cos (s- \frac{\pi \nu}{2} - \frac{\pi}{4}) + O(s^{-3/2}) as s \to \infty.

(Prop 1.2.) Suppose a and m are fixed, with a >0 and m > -1. Then as s \to \infty \int_{0}^{a} e^{-sx} x^{m} dx = s^{-m-1} \Gamma(m+1) + O(e^{-cs}), for some positive c.

(Prop 1.3.) Suppose a and m are fixed, with a >0 and -1<m<0. Then as |s| \to \infty with Re(s) \ge 0, \int_{0}^{a} e^{-sx} x^{m} dx = s^{-m-1} \Gamma(m+1) + O(1/|s|). (Here s^{-m-1} is the branch of that function that is positive for s >0.)

(subsection 2) Laplace’s method ; Stirling’s formula

(Def) Consider \int_{a}^{b} e^{-s\Phi(x)} \psi(x) dx where the phase \Phi is real-valued, and both it and the amplitude \psi are assumed for simplicity to be indefinitely differentiable. Our hypothesis regarding the minimum of \Phi is that there is an x_{0} \in (a,b) so that \Phi’(x_{0}) = 0, but \Phi”(x_{0}) >0 throughout [a,b].

(Prop 2.1.) Under the above assumptions, with s >0 and s \to \infty, \int_{a}^{b} e^{-s\Phi(x)}\psi(x) dx = e^{-s\Phi(x_{0})} [\frac{A}{s^{1/2}} + O(\frac{1}{s})], where A = \sqrt(2\pi) \frac{\psi(x_{0})}{(\Phi”(x_{0})^{1/2}}.

(Prop 2.2.) With the same assumptions on \Phi and \psi, the relation of (prop 2.1.) continues to hold if |s| \to \infty with Re(s) \ge 0.

(Def) The special case of (Prop 2.2.) When s is purely imaginary, s = it, t \to \mp \infty , is often treated separately; the argument in this situation is usually referred to as the method of stationary phase. The points x_{0} for which \Phi’(x_{0}) = 0 are called the critical points.

(Thm 2.3.) If |s| \to \infty with s \in S_{\delta} , then \Gamma(s) = e^{s log s} e^{-s} \frac{\sqrt{2\pi}}{s^{1/2}} (1+ O(\frac{1}{|s|^{1/2}}) ).

(subsection 3) The Airy function

(Def) The Airy function Ai is defined by Ai(s) = \frac{1}{2\pi} \int_{-\infty}^{\infty} e^{i(x^{3}/3+sx)}dx , with s \in \mathbb{R}.

Ai”(s) = s Ai(s).

(Thm 3.1.) Suppose u >0. Then as u \to \infty,

Ai(-u) = \pi^{-1/2} u^{-1/4} cos(\frac{2}{3} u^{3/2} - \frac{\pi}{4})(1+O(1/u^{3/4})).

Ai(u) = \frac{1}{2\pi^{1/2}} u^{-1/4} e^{-\frac{2}{3} u^{3/2}} (1+ O(1/u^{3/4})).

(De) Assume F(z) is holomorphic. We seek a contour \Gamma so that

Im(F) = 0 on \Gamma.

Re(F) has a minimum on \Gamma at some point z_{0}, and this function is non-degenerate in the sense that the second derivative of Re(F) along \Gamma is strictly positive at z_{0}.

If as above, F”(z_{0}) \neq 0, then there are two curves \Gamma_{1} and \Gamma_{2} passing through z_{0} which are orthogonal, so that F|{\Gamma}, is real for i = 1,2, with Re(F) restricted to \Gamma{1} having a minimum at z_{0}; and Re(F) restricted to \Gamma_{2} having a maximum at z_{0} . We try to deform out original contour of integration to \Gamma = \Gamma_{1}. This approach is usually referred to as the method of steepest descent.

(subsection 4) The partition function

(Thm 4.1.) If p denotes the partition function, then

p(n)~ \frac{1}{4\sqrt{3} n} e^{Kn^{1/2}} as n \to \infty, where K = \pi \sqrt{\frac{2}{3}}.

A much more precise assertion is that p(n) = \frac{1}{2\pi \sqrt{2}} \frac{d}{dn} (\frac{e^{K(n-\frac{1}{24})^{1/2}}}{(n-\frac{1}{24})^{1/2}}) + O(e^{\frac{K}{2} n^{1/2}}).

(Appendix B) Simple Connectivity and Jordan Curve Theorem

(subsection 1) Equivalent formulations of simple connectivity

(Thm 1.1.) A region \Omega is holomorphically simply connected iff \Omega is simply connected.

(Thm 1.2.) If \Omega is a bounded region in \mathbb{C}, then \Omega is simply connected iff the complement of \Omega is connected.

(Def) The winding number of a closed curve \gamma around a point z \notin \gamma is W_{\gamm}(z) = \frac{1}{2\pi i} \int_{\gamma} \frac{d\zeta}{\zeta - z}. Sometimes, W_{\gamma}(z) is also called the index of z with respect to \gamma.

(Lem 1.3.) Let \gamma be a closed curve in \mathbb{C}.

If z \notin \gamma, then W_{\gamma}(z) \in \mathbb{Z}.

If z and w belong to the same open connected component in the complement of \gamma, then W_{\gamma}(z) = W_{\gamma}(w).

If z belongs to the unbounded connected component in the complement of \gamma, then W_{\gamma}(z) = 0.

(Thm 1.4.) A bounded region \Omega is simply connected iff W_{\gamma}(z) = 0 for any closed curve \gamma in \Omega and any point z not in \Omega.

(Lem 1.5.) Let w be any point in F_{1}. Under the above assumptions. there exists a finite collection of closed squares \mathcal{Q} = {Q_{1},…,Q_{n}} that belong to a uniform grid \mathcal{G} of the plane, and are such that:

w belongs to the interior of Q_{1}.

The interiors of Q_{j} and Q_{k} are disjoint when j \neq k.

F_{1} is contained in the interior of \cup_{j=1}^{n} Q_{j}.

\cup_{j=1}^{n} Q_{j} is disjoint from F_{2}.

The boundary of \cup_{j=1}^{n} Q_{j} lies entirely in \Omega, and consists of a finite union of disjoint simple closed polygonal curves.

(subsection 2) The Jordan curve theorem

(Thm 2.1.) Let \Gamma be curve in the plane that is simple and piecewise smooth. Then, the complement of \Gamma is an open connected set whose boundary is precisely \Gamma.

(Thm 2.2.) Let \Gamma be a curve in the plane which is simple, closed, and piecewise-smooth. Then, the complement of \Gamma consists of two disjoint connected open sets. Precisely one of these regions is bounded and simply connected; it is called the interior of \Gamma and denoted by \Omega. The other component is unbounded, called the exterior of \Gamma, and denoted by \mathcal{U}. Moreover, with the appropriate orientation for \Gamma, we have W_{\Gamma}(z) = \begin{cases} 1 & if z \in \Omega \ 0 & if z \in \mathcal{U}\end{cases}.

(Thm 2.3.) Suppose f is a function that is holomotphic in the interior \Omega of a simple closed curve \Gamma. Then \int_{\eta} f(\zeta) d \zeta = 0 whenever \eta is any closed curve contained in \Omega.

(prop) recall that an arc-length parametrization \gamma for a smooth curve \Gamma_{0} satisfies |\gamma’(t)| = 1 for all t. Every smooth curve has an arc-length parametrization.

(Lem 2.4.) Let \Gamma_{0} be a simple smooth curve with an arc-length parametrization given by \gamma : [0,L] \to \mathbb{C}. For each real number \epsilon, Let \Gamma_{\epsilon} be the continuous curve defined by the parametrization \gamma_{\epsilon} (t) = \gamma(t) + i\epsilon \gamma’(t) , for 0 \le t \le L. Then, there exists \kappa_{1}>0 so that \Gamma_{0} \cap \Gamma_{\epsilon} = \empty whenever 0 < |\epsilon| < \kappa_{1}.

(Lem 2.5.) Suppose z is a point which does not belong to the smooth curve \Gamma_{0}, but that is closer to an interior point of the curve than to either of its end-points. Then z belongs to \Gamma_{\epsilon} for some \epsilon \neq 0. More precisely, if z_{0} \in \Gamma_{0} is closest to z and z_{0} = \gamma(t_{0}) for some t_{0} in the open interval (0,L) , then z = \gamma(t_{0}) + i\epsilon \gamma’(t_{0}) for some \epsilon \neq 0.

(prop 2.6.) Let A and B denote the two end-points of a simple smooth curve \Gamma_{0}, and suppose that K is a compact set that satisfies either \Gamma_{0} \cap K = \empty or \Gamma_{0} \cap K = A \cup B. If z \notin \Gamma_{0} and w \notin \Gamma_{0} lie on the same side of \Gamma_{0}, and are closer to interor points of \Gamma_{0} than they are to K or to the end-points of \Gamma_{0}, then z and w can be joined by a continuous curve that lies entirely in the complement of K \cup \Gamma_{0}.

(Lem 2.7.) Let \Gamma_{0} be a simple smooth curve. There exists \kappa_{2} >0 so that the set N, which consists of points of the form z = \gamma(L) + \epsilon e^{i\theta} \gamma’(L) with -\pi/2 \ge \theta \ge \pi/2 and 0 < \epsilon < \kappa_{2} , is disjoint from \Gamma_{0}.

(Prop 2.8.) Let A denote an end-point of the simple smooth curve \Gamma_{0}, and suppose that K is a compact set that satisfies either \Gamma_{0} \cap K = \empty or \Gamma_{0} \cap K = A. If z \notin \Gamma_{0} and w \notin \Gamma_{0} are closer to inerior points of \Gamma_{0} than they are to K or to the end-points of \Gamma_{0}, then z and w can be joined by a continuous curve that lies entirely in the complement of \Gamma_{0} \cap K.

(Thm 2.9.) If a function f is holomorphic in an open set that contains a simple closed piecewise-smooth curve \Gamma and its interior, then \int_{\Gamma} f = 0.

(Lem 2.10) Let \gamma : [0,1] \to \mathbb{C} be a simple smooth curve. Then, for all sufficiently small \delta >0 the circle C_{\delta} centered at \gamma(0) and of radius \delta intersects \gamma in precisely one point.

Real Analysis

르벡적분

(section 1) 르벡적분

(subsection 1.1) 잴 수 있는 집합과 측도

(정의) 열린 집합 U \mathbb{R}의 한 점 x \in U 를 고정하고 a_{x} = inf{a\in \mathbb{R} : (a,x) \subseteq U } , b_{x} = sup{b\in \mathbb{R} : (a,b) \subseteq U } 라 정의한다. I_{x} = (a_{x} , b_{x}) 라 두자. U = \bigcup_{x \in U} I_{x} 는 서로소인 열린 구간들의 합집합이다.

(명제 1.1.1) 실수의 부분집합 U \subseteq \mathbb{R} 가 열린 집합이라면, U는 셀 수 있는 서로소인 열린 구간의 합집합으로 표시된다. 또한 이 표시방법은 한 가지 뿐이다.

(정의) 열린 집합 U \mathbb{R} 가 서로소인 열린 구간의 합집합 \sqcup_{n}(a_{n} b_{n}) 으로 표시되었을 때 그 길이 \lambda(U) 를 \lambda(U) = \sum_{n = 1}^{\infty} (b_{n} – a_{n}) 으로 정의한다.

(정의) 임의의 집합 S \subset \mathbb{R} 에 대하여 그 길이 \mu(S) 를 \mu(S) = inf{\lambda(U) : U \supseteq S, U는 열린 집합} 으로 정의한다.

(정의) 집합 S \subset \mathbb{R}과 점 x \in \mathbb{R} 에 대하여 x + S = {x + y \in \mathbb{R} : y \in S} 로 정의하고 이를 집합 S 의 평행이동이라 부른다.

(명제 1.1.2) 함수 \mu : \mathcal{P}(R) \to [0,\infty] 는 다음 성질을 가진다.

만일 S \subseteq T 이면 \mu(S) \le \mu(T) 이다.

만일 I 가 a, b를 양끝점으로 하는 구간이면 \mu(I) = |b – a| 이다.

임의의 x \in \mathbb{R} 와 S \ini \mathcal{R} 에 대하여 \mu(x + S) = \mu(S)이다.

임의의 셀 수 있는 집합 모임 {S_{n} : n = 1,2,…} 에 대하여 \mu \biggl( \bigcup_{n = 1}^{\infty} S_{n} \biggr) \le \sum_{n=1}^{\infty} \mu(S_{n}) 이 성립한다.

(정의) 집합 E \subseteq \mathbb{R}이 다음 조건 A \subseteq \mathbb{R} \implies \mu(A) = \mu(A \cap E) + \mu(A \cap E^{c}) 를 만족하면 E를 잴 수 있는 집합이라 부르고, 잴 수 있는 집합 전체의 모임을 \mathfrak{M} 이라 쓴다.. 잴 수 있는 집합의 조건은 카라테오도리 조건이라 불린다.

(정리 1.1.3) 집합 모임 \mathfrak{M}은 다음 성질을 가진다.

\empty \subset \mathfrak{M}

만일 E \in \mathfrak{M} 이면 E^{c} \in \mathfrak{M} 이다.

만일 E, F \in \mathfrak{M} 이면 E \cup F \in \mathfrak{M} 이다.

각 n = 1,2, … 에 대하여 E_{n} \in \mathfrak{M} 이면 \bigcup_{n} E_[n] \in \mathfrak{M} 이다.

만일 \mu€ = 0 이면 E \in \mathfrak{M} 이다.

집합 I 가 구간이면 I \in \mathfrak{M} 이다.

집합 S가 열린 집합이면 S \in \mathfrak{M} 이다.

(정리 1.1.4) 각 n \ 1,2, … 에 대하여 E_{n} \in \mathfrak{M} 이고 집합모임 {E_{n}} 이 서로소이면 등식 \mu \biggl( \bigsqcup_{n = 1}^{\infty} E_{n} \biggr) = \sum_{n=1}^{\infty} \mu(E_{n}) 이 성립한다.

(따름정리 1.1.5) 잴 수 있는 단순증가 집합열 <E_{n}> 에 대하여 등식 \mu \biggl( \bigcup_{n = 1}^{\infty} E_{n} \biggr) = \lim_{n \to \infty} \mu(E_{n}) 이 성립한다. 또한 잴 수 있는 단순감소 집합열 <F_{n}> 에 대하여 \mu(F_{1}) < \infty 이면 등식 \mu \biggl( \bigcap_{n = 1}^{\infty} F_{n} \biggr) = \lim_{n \to \infty} \mu(F_{n}) 이 성립한다.

(정의) 열린 집합들의 셀 수 있는 교집합으로 표시되는 집합을 G_{delta} -집합 이라 하고, 닫힌 집합들의 셀 수 있는 합집합으로 표시되는 집합을 F_{delta} -집합이라 부른다. 이 두 부류의 집합들은 잴 수 있다.

(명제 1.1.6) 집합 E \subset \mathbb{R} 에 대하여 다음이 동치이다.

E가 잴 수 있는 집합이다.

임의의 양수 \epsilon > 0 에 대하여 F \subset E \subset U, \mu(U \setminus F) < \epsilon 인 열린 집합 U 와 닫힌 집합 F 가 존재한다.

B \subset E \subset A , \mu(A \setminus B) = 0 를 만족하는 G_{delta}= 집합 A 와 F_{delta} -집합 B 가 존재한다.

(따름정리 1.1.7) 임의의 잴 수 있는 집합 E \subset \mathbb{R} 에 대하여 등식 \mu(E) = \sup{\mu(K) : K \subset E, K 는 닫힌 유계집합}

(정의) 우선 구간 I = [0,1] 을 삼등분하여 가운데 열린구간 J_{1} 를 들어내고, 즉 C_{1} = I \setminus J_{1} 이라 두고 C_{1}의 두 구간에 대하여도 똑 같은 일을 되풀이한다. C_{2} 의 네 구간에 대하여도 똑 같은 일을 되풀이한다. 이렇게 하여 얻은 닫힌 집합열 \langle C_{n} \rangle 들의 교집합을 C = \bigcup C_{n} 이라 두고 이를 칸토르 집합이라 한다.

(정의) 함수 \mu : \mathcal{P} (\mathbb{R}) \to [0,\infty] 를 \mathfrak{M} 에 제한했을 때 이를 m : \mathfrak{M} \to [0,\infty] 이라 쓰자.

(정의) 임의의 집합 X의 멱집합의 부분집합 \mathfrak{S} 가 (정리 1.1.3) 의 첫 번쨰 조건부터 네 번째 조건까지를 만족하면, 이를 \sigma – 대수라 부른다.

(정의) X의 \sigma – 대수가 주어지고 함수 \mu : \mathfrak{S} \to [0,\infty] 가 조건 \mu \biggl( \bigsqcup_{n = 1}^{\infty} E_{n} \biggr) = \lim_{n \to \infty} \mu(E_{n}) 를 만족하면, (\mathfrak{S}, \mu) 를 X의 측도라 하고, \mathfrak{S} 의 원들을 잴 수 있는 집합이라 부른다.

따라서 (\mathfrak{M},m) 은 \mathbb{R}의 측도인데, 이를 르벡 측도라 한다.

(정의) 집합 X의 한 점 x \in X를 고정하고, 임의의 집합 A \subset X 에 대하여 \delta_{x}(A) = \begin{cases}1 & x \in A \ 0 & x \notin A \end{cases} 라 정의하면 (\mathcal{P}(X) , \delta_{x}) 는 X의 측도가 되는데, 이렇게 정의된 측도를 디락 측도라 부른다.

(subsection 1.2)잴 수 있는 함수와 적분

(정의) 확장실수체 \mathbb{R}^{*} = \mathbb{R} \cup {\mp \infty}

(정의) 잴 수 있는 집합 E \subseteq \mathbb{R} 에서 정의된 함수 f : E \to \mathbb{R}^{*} 이 다음 조건 a\in \mathbb{R} \implies {x \in E : f(x) > a} \in \mathfrak{M} 을 만족하면 이를 잴 수 있는 함수라 한다.

(정의) 집합 A \subseteq \mathbb{R} 에 대하여, 그 특성함수 \chi_{A} : \mathbb{R} \to \mathbb{R} 를 \chi_{A}(x) = \begin{cases}1 & x \in A \ 0 & x \notin A \end{cases} 라 정의하자.

그러면 집합 A가 잴 수 있을 필요충분조건은 \chi_{A} 가 잴 수 있는 함수임이다.

(명제 1.2.1) 잴 수 있는 집합 E 에서 정의된 함수 f : E \to \mathbb{R}^{*} 에 대하여 다음은 동치이다.

함수 f는 잴 수 있는 함수이다.

임의의 a \in \mathbb{R} 에 대하여 {x \in E : f(x) \ge a} \in \mathfrak{M} 이다.

임의의 a \in \mathbb{R} 에 대하여 {x \in E : f(x) < a} \in \mathfrak{M} 이다.

임의의 a \in \mathbb{R} 에 대하여 {x \in E : f(x) \le a} \in \mathfrak{M} 이다.

(따름정리 1.2.2) 함수 f: E \to \mathbb{R}^{} 이 잴 수 있으면, 임의의 a \in \mathbb{R}^{} 에 대하여 집합 {x \in E : f(x) = a} 는 잴 수 있다.

(명제 1.2.3) 잴 수 있는 집합 E 에서 \mathbb{R}^{*} 로 가는 함수들에 대하여 다음이 성립한다.

연속함수 f : E \to \mathbb{R} 은 잴 수 있다.

만일 f, g가 잴 수 있고 \alpha \in \mathbb{R} 이면 f + g, fg, \alpha f 도 잴 수 있다.

각 n = 1,2, …. 에 대하여 f_{n} 이 잴 수 있으면 함수 x \mapsto \inf_{n} f_{n} (x) 와 x \mapsto \sup_{n}f_{n}(x) 도 잴 수 있다.

각 n = 1,2, …. 에 대하여 함수 f_{n} 이 잴 수 있으면 점별극한함수 \limsup_{n} f_{n} , \liminf_{n} f_{n}, \lim_{n} f_{n} 도 잴 수 있다.

함수 f : E \to \mathbb{R}^{*} 이 잴 수 있고, g : \mathbb{R} \to \mathbb{R} 이 연속이면 그 합성함수 g \bullet f 도 잴 수 있다. 여기서, f(x) = \mp \infty 인 x \in E 에 대해서는 (g \bullet f) (x) = \mp \infty 로 정의한다.

(정의) 잴 수 있는 함수 f : E \to \mathbb{R} 의 치역이 유한집합이면, 이를 단순함수라 한다.

치역이 {c_{1}, …, c_{n}} 인 단순함수 s : E \to \mathbb{R} 가 있을 때, 각 i = 1,2,…, n 에 대하여 E_{i} = { x \in E : s(x) = c_{i}} 라 두면 {E_{i}} 는 서로소이고 s 는 s = \sum_{i = 1}^{n} c_{i} \chi_{E_{i}} 같이 잴 수 있는 특성함수들의 선형결합으로 표시할 수 있다.

(정의) 정의된 단순함수가 s \ge 0 일때, 그 적분을 \int_{E} s = \sum_{i = 1}^{n} c_{i} m(E_{i}) 라 정의한다.

\int_{E} (s+t) = \int_{E} s + \int_{E} t

\int_{E} (cs) = c\int_{E} s

s \le t \implies \int_{E} s \le \int_{E} t

(정의) 잴 수 있는 함수 f : E \to [0, +\infty] 의 적분을 \infty_{E} f = sup{\infty_{E} s : 0 \le s \le f, s 는 단순함수} 로 정의한다.

0 \le g \le f \implies 0 \le \int g \le \int f

(정리 1.2.4) 임의의 잴 수 있는 함수 f : E \to [0,\infty] 는 단조증가하는 단순함수의 점별극한함수로 표시된다.

(따름정리 1.2.5) 임의의 잴 수 있는 함수 f : E \to [0, +\infty] 는, 적절한 양수열 \left \langle c_{n} \right \rangle 과 잴 수 있는 집합열 \left \langle E_{n} \right \rangle 에 대하여 f = \sum_{n = 1}^{\infty} c_{n} \chi_{E_{i}} 로 표시된다.

(정의) 함수 f : X \to \mathbb{R}^{} 에 대하여 f_{+} (x) = max{f(x),0}, f_{-} (x) = - min{f(x),0}, x \in X 라 두면 f_{+}, f_{-} \ge 0 이고 f = f_{+} – f_{-} 이다. 만일 함수 f : E \to \mathbb{R}^{} 가 잴 수 있으면 (명제 1.2.3)에 의하여 f_{+} 와 f_{-} 도 잴 수 있다.

(정의) 함수 f 의 적분을 \int_{E} f = \int_{E} f_{+} - \int_{E} f_{-} 라 정의한다. 만일 함수 f_{+} 와 함수 f_{-} 의 적분값이 모두 무한이면 f의 적분을 정의하지 않는다.

(정의) 함수 f_{+}와 f_{-} 의 적분값이 모두 유한일 때, 함수 f 를 르벡적분 가능한 함수 혹은 L^{1}-함수라 부른다. 또한, 잴수 있는 집합 E 위에서 정의된 르벡적분 가능한 함수 전체의 집합을 L^{1} (E) 라 쓴다.

만일 f \le g 이면 \iint_{E} f \le \int_{E} g 이다.

만일 m(E) = 0 이면, 임의의 함수 f : E \to \mathbb{R}^{*} 가 잴 수 있고 \int_{E} f = 0 이다.

(정리 1.2.6) 서로소인 잴 수 있는 집합 모임 {A_{n} : n = 1,2, …} 에 대하여 A = \sqcup_{n = 1}^{\infty} A_{n} 이라 하자. 그러면 임의의 잴 수 있는 함수 f : A \to [0,\infty] 에 대하여 등식 \infty_{A} f = \sum_{n = 1}^{\infty} \int_{A_{n}} f 이 성립한다.

(따름정리 1.2.7) 잴 수 있는 함수 f 에 대하여 f \in L^{1} (E) 일 필요충분조건은 |f| \in L^{1} (E) 이고 이 때 부등식 |\int_{E} f| \le \int_{E} |f| 이 성립한다.

(정의) 잴 수 있는 집합 E 의 각 점 x \in E 에 관한 명제 P(x) 가 있을 때, m({x \in E : P(x) 가 성립하지 않는다}) = 0 이면 거의 모든 점에서 P(x) 가 성립한다고 한다.

(명제 1.2.8) 잴 수 있는 함수 f : E \to [0,\infty] 의 적분값이 유한이면 f는 거의 모든 점에서 유한값을 가진다.

(정의) 잴 수 있는 집합 E에서 정의된 복소함수 f : E \to \mathbb{C} 에 대해 함수 f = u + iv 의 실수부를 u, 허수부를 v라 하자. u와 v가 잴 수 있으면 f가 잴 수 있다고 한다.

만일 \int_{E} |f| 가 유한값이면 f를 르벡적분가능함수라 부른다.

(정의) 복소함수의 적분을 \int_{E} (u + iv) = \int_{E} u + i \int_{E} v 라 정의한다. 앞으로 L^{1}(E) 는 E에서 \mathbb{C} 로 가는 르벡적분가능함수 전체의 집합을 나타낸다.

L^{1}(E) 는 벡터공간이다.

\int_{E} (\alpha f) = \alpha \int_{E} f f \in L^{1}(E), \alpha \in \mathbb{C}

|\int_{E} f| \le \int_{E} |f|

(따름정리 1.2.9) 서로소인 잴 수 있는 집합모임 {A_{n} : n = 1,2,…} 에 대하여 A = \sqcup_{n = 1}^{\infty} A_{n} 이라 하자. 그러면 임의의 f \in L^{1}(A) 에 대하여 \infty_{A} f = \sum_{n = 1}^{\infty} \int_{A_{n}} f 이 성립한다.

(명제 1.2.10) 잴 수 있는 집합 E에서 정의된 잴 수 있는 함수 f 에 대하여 다음이 성립한다.

만일 f \ge 0 이고 \inf_{E} f = 0 이면 거의 모든 점에서 f = 0 이다.

만일 f \in L^{1} (E) 이고 임의의 잴 수 있는 집합 D \subset E 에 대하여 \int_{D} f = 0 이면 거의 모든 점에서 f = 0이다.

(명제 1.2.11) 구간 [a,b] 위에서 함수 f : [a,b] \to \mathbb{R} 가 잴 수 있고, 모든 점 x 에 대하여 등식 \int_{[a,x]} f = 0 이 성립하면, 거의 모든 점에서 f = 0 이다.

(subsection 1.3) 적분의 수렴정리

(단조수렴정리)(정리 1.3.1) 잴 수 있는 집합 E에서 \mathbb{R}^{*} 로 가는 잴 수 있는 함수열 \left \langle f_{n} \right \rangle 이 다음 조건 0 \le f_{n} \le f_{n+1}, n = 1, 2, … 을 만족한다고 하자. 만일 각 x \in E 에 대하여 f(x) = lim_{n} f_{n}(x) 이면, \inf_{E} f = \lim_{n \to \infty} \int_{E} f_{n} 이 성립한다.

(정리 1.3.2) 잴 수 있는 집합 E에서 정의된 잴 수 있는 함수 f, g 에 대하여 다음이 성립한다.

만일 f, g \in L^{1}(E) 이면 f + g \in L^{1} (E) 이다. 또한 f, g \ge 0 이거나 f, g \in L^{1} (E) 이면 등식 \int_{E} (f + g) = \int_{E} f + \int_{E} g 이 성립한다.

만일 f \in L^{1}(E) 이고 \alpha \in \mathbb{C} 이면, \alpha f \in L^{1} (E) 이고 등식 \int_{E} (\alpha f) = \alpha \int_{E} f 이 성립한다.

(따름정리 1.3.3) 잴 수 있는 집합 E에서 [0,\infty] 로 가는 잴 수 있는 함수열 \left \langle f_{n} \right \rangle 에 대하여 f(x) = \sum_{n = 1}^{\infty} f_{n}(x) 라 정의하면, 등식 \int_{E} f = \sum_{n = 1}^{\infty} \int_{E} f_{n} 이 성립한다.

(파투) (정리 1.3.4) 잴 수 있는 집합 E에서 [0, + \infty] 로 가는 잴 수 있는 함수열 \left \langle f_{n} \right \rangle 에 대하여, 부등식 \int_{E} (\liminf_{n \to \infty} f_{n}) \le \liminf_{n \to \infty} \int_{E} f_{n} 이 성립한다.

(르벡 수렴정리)(정리 1.3.5) 잴 수 있는 집합 E 에서 \mathbb{C} 로 가는 잴 수 있는 함수열 \left \langle f_{n} \right \rangle 이 다음 조건을 만족한다고 하자.

각 n = 1,2, … 에 대하여 |f_{n}| \le g 인 g \in L^{1}(E) 가 존재한다.

각 x \in E 에 대하여 극한값 \lim_{n} f_{n}(x) 가 존재한다.

이 때, f(x) = lim_{n} f_{n}(x) 라 두면, f \in L^{1}(E) 이고 등식 \lim_{n \to \infty} \int_{E} |f_{n} – f| = 0, \lim_{n \to \infty} \int_{E} f_{n} = \int_{E} f 이 성립한다.

(레비)(정리 1.3.6) 잴 수 있는 집합 E 에서 \mathbb{C} 로 가는 L^{1}-함수열 \left \langle f_{n} \right \rangle 에 대하여 \sum_{n = 1}^{\infty} \int_{E} |f_{n}| < \infty 이라 가정하자. 그러면 거의 모든 x \in E 에 대하여 급수 \sum_{n = 1}^{\infty} f_{n}(x) 가 절대수렴한다. 또한, 그 극한값을 f(x) = \sum_{n = 1}^{\infty} f_{n}(x) 라 두면 f \in L^{1}(E) 이고 그 적분값은 \int_{E} f = \sum_{n = 1}^{\infty} \int_{E} f_{n} 로 주어진다.

(명제 1.3.7) 열린구간 I, J 의 곱집합 I \times J 위에서 정의된 이변수함수 f : I \times J \to \mathbb{C} 가 주어져있다. 각 x \in I 에 대하여 일변수함수 y \mapsto f(x,y) 가 잴 수 있는 함수이고, I \times J 위에서 편도함수 D_{1}f 가 존재한다고 하자. 만일 조건 |D_{1}f(x,y)| \le g(y), x \in I 를 만족하는 적분가능함수 g \in L^{1}(J) 가 존재하면 등식 \frac{d}{dx} \int_{J} f(x,y) dy = \int_{J} D_{1}f(x,y)dy 이 성립한다.

(명제) 세 조건을 만족하는 함수 \mu : \mathcal{P}(\mathbb{R}) \to [0,\infty] 는 존재하지 않는다.

만일 I 가 a, b를 양끝점으로 하는 구간이면 \mu(I) = |b – a| 이다.

임의의 x \in \mathbb{R} 와 S \ini \mathcal{R} 에 대하여 \mu(x + S) = \mu(S)이다.

\biggl( \bigsqcup_{n = 1}^{\infty} E_{n} \biggr) = \lim_{n \to \infty} \mu(E_{n})

(명제 1.3.6) 집합 E \subset \mathbb{R} 의 모든 부분집합이 잴 수 있으면 \mu(E) = 0 이다.

(subsection 1.4) 리만적분과 르벡적분

(정의) 유계함수 f : [a,b] \to [0, \infty] 와 분할 P = {x_{0}(=a) , x_{1}, …, x_{n} (=b)} 및 i = 1,2, …, n 에 대하여 M_{i} = \sup{f(x) : x_{i-1} \le x \le x_{i}}, m_{i} = \inf{f(x) : x_{i-1} \le x \le x_{i}} 라 두고, 두 단순함수 U_{p} 와 L_{p} 를 U_{p} = f(a){ \chi{{a}}} + \sum_{i = 1}^{n} M_{i\chi_{(x_{i-1}, x_{i}\rbrack}}, L_{p} = f(a){ \chi{{a}}} + \sum_{i = 1}^{n} m_{i\chi_{(x_{i-1}, x_{i}\rbrack}}

(정의)상합과 하합을 U_{a}^{b} (f,P) = \int_{[a,b]} U_{p} , L_{a}^{b} (f,P) = \int_{[a,b]} L_{p} 와 같이 쓸 수 있다. 구간 [a,b]의 분할 전체의 집합을 \mathcal{P}[a,b] 라 두고 f의 리만 상적분과 리만 하적분을 각각 \overline{\int_{a}^{b} f }= \inf{ U_{a}^{b} (f,P) : P \in \mathcal{P}[a,b] }, \underline{ \int_{a}^{b} f }= \sup{ L_{a}^{b} (f,P) : P \in \mathcal{P}[a,b] } 와 같이 정의한다.

(정의) \lVert P \rVert := max{|y_{i} – y_{i-1}| : i = 1,2,…,m} 이라 정의하고 \lVert P_{n} \rVert 이 되도록 분할열 \langle P_{n} \rangle 을 잡으면 lim_{n \to \infty} U(f,P_{n}) = \overline{\int_{a}^{b} f } 이다. lim_{n \to \infty} L(f,P_{n}) = \underline{\int_{a}^{b} f } 이다. 만일 \underline{\int_{a}^{b} f } = \overline{\int_{a}^{b} f } 이면 f가 리만적분가능하다고 하고 그 공통값을 \int_{a}^{b} f 라 쓴다.

(정리 1.4.1) 유계함수 f : [a,b] \to \mathbb{R} 에 대하여 다음이 성립한다.

만일 f 가 리만적분가능하면 f 는 잴 수 있고, 등식 \int_{[a,b]} f = \int_{a}^{b} f 이 성립한다.

함수 f 가 리만적분가능할 필요충분조건은 거의 모든 점에서 연속임이다.

미분과 르벡적분

(section 2) 미분과 르벡적분

(subsection 2.1) 단조함수의 미분과 적분의 미분

적분가능함수 f : [a,b] \to \mathbb{R} 에 대하여 새로운 함수 F를 다음 F(x) = \int_{a}^{b} f, x \in [a,b] 와 같이 정의한다.

(비탈리)(도움정리 2.1.1) 잴 수 있는 집합 E \subset \mathbb{R} 이 \mu(E) < \infty 이고, 닫힌구간모임 \mathcal{I} 가 다음조건

임의의 \epsilon > 0 과 x \in E 에 대하여 x \in I 및 \mu(I) < \epsilon 을 만족하는 구간 I \in \mathcal{I} 가 존재한다.

을 만족하면, 임의의 양수 \epsilon >0 에 대하여 \mu[E \setminus (I_{1} \sqcup I_{2} \sqcup … \sqcup I_{n})] < \epsilon 이 성립하는 유한개의 서로소인 부분모임 {I_{1}, I_{2}, …, I_{n}} \subset \mathcal{I} 이 존재한다.

(정의) 열린구간 I 에서 정의된 함수 f : I \to \mathbb{R} 의 정의역의 한 점 x \in I 에 대하여 다음 값들을 정의하자. 이 값들은 \mp \infty 를 취할 수 있는데, 이를 디니 미분계수라 한다.

D^{+} f(x) = \limsup_{h \to 0+} \frac{f(x+h) – f(x)}{h}

D_{+} f(x) = \liminf_{h \to 0+} \frac{f(x+h) – f(x)}{h}

D^{-} f(x) = \limsup_{h \to 0+} \frac{f(x) – f(x-h)}{h}

D_{-} f(x) = \liminf_{h \to 0+} \frac{f(x) – f(x-h)}{h}

(명제) 함수 f 가 x 에서 미분가능할 필요충분조건은 디니미분계수가 모두 유한이고 일치함이다.

(르벡)(정리 2.1.2) 임의의 단조증가함수 f : [a,b] \to \mathbb{R} 은 거의 모든 점에서 미분가능하다.

(명제 2.1.3) 단조증가함수 f : [a,b] \to \mathbb{R} 의 도함수 f’ 은 잴 수 있는 함수이고 부등식 \int_{a}^{b} f’ \le f(b) – f(a) 를 만족한다.

(명제 2.1.4) 잴 수 있는 집합 E에서 정의된 L^{1} 함수 f \in L^{1}(E) 가 주어져 있을 때, 임의의 양수 \epsilon >0 에 대하여 성질 A \in \mathfrak{M}, m(A) < \delta \implies \int_{A} |f| < \epsilon 을 만족하는 양수 \delta >0 가 존재한다.

(첫 번째 미적분학의 기본정리) (정리 2.1.5) 르벡적분가능한 함수 f : [a,b] \to \mathbb{R} 에 대하여 F(x) = F(a) + \int_{a}^{x} f, x \in [a,b] 라 정의하면, 거의 모든 점에서 F’ = f 가 성립한다.

(푸비니) (정리 2.1.6) 구간 [a,b] 에서 정의된 단조증가함수로 이루어진 급수 \sum_{n} f_{n} 이 s 로 점별수렴하면 거의 모든 점에서 s’ = \sum_{n} f’_{n} 이 성립한다.

(subsection 2.2) 절대연속함수와 미분의 적분

(정의)칸토르 집합에 대하여, 각 n = 1,2,… 에 대하여 g(n) = \frac{3^{n}}{2^{n}}\chi_{C_{n}} , f_{n}(x) = \int_{0}^{x} g_{n}, x \in [0,1] 이라 정의하자. |f_{n}(x) – f_{n+1}(x)| \le \frac{1}{2^{n-1}} 이므로, 함수열 \langle f_{n} \rangle 은 단조증가 연속함수로 고르게 수렴하는데, 그 극한함수 \phi 를 칸토르함수라 한다.

\phi는 거의 모든점에서 미분가능하고 \phi’ 이 르벡적분가능함에도 불구하고 등식 int_{a}^{x} f’ = f(x) – f(a) , x \in [a,b] 이 성립하지 않는다.

(정의) 함수 f : [a,b] \to \mathbb{C} 가 주어져 있을 때, 임의의 \epsilon >0 에 대하여 성질

{(a_{i},b_{i} : i = 1,2,…,n} 이 서로소인 [a,b] 의 부분구간모임이고 \sum_{i = 1}^{n} (b_{i} – a_{i}) < \delta 이면 \sum_{i = 1}^{n}|f(b_{i}) – f(a_{i})| < \epsilon 이다.

을 만족하는 양수 \delta > 0 가 존재하면 f 를 절대연속함수라 한다.

(명제) f 가 적분가능하면 F는 절대연속이다. 절대연속함수는 고른연속함수이지만, 그 역은 성립하지 않는다.

(정의) 함수 f : [a,b] \to \mathbb{C} 와 구간 [a,b] 의 분할 P = {x_{0}, … , x_{n}} 가 주어졌을 때, 분할 P 에 의한 함수 f의 변동 V_{a}^{b} (f,P) 를 V_{a}^{b} (f,P) = \sum_{i=1}^{n} |f(x_{i}) – f(x_{i-1})| 로 정의한다.

(정의) 집합 { V_{a}^{b} (f,P): P \in \mathcal{P}[a,b]} 가 위로 유계일 때, 함수 f 를 유계변동함수라 하고, 이 경우 전변동 V_{a}^{b} (f) 를 V_{a}^{b} (f) = \sup{V_{a}^{b} (f,P) : P \in \mathcal{P}[a,b]} 라 정의한다.

(죠르당)(정리 2.2.1) 함수 f : [a,b] \to \mathbb{R} 가 유계변동일 필요충분조건은 f가 단조증가함수의 차로 표시됨이다.

따라서 유계닫힌구간에서 정의된 유계변동함수는 거의 모든 점에서 미분가능하다.

(명제 2.2.2) 모든 절대연속함수 f : [a,b] \to \mathbb{C} 는 유계변동함수이다.

따라서 유계닫힌구간에서 정의된 임의의 절대연속함수가 거의 모든 점에서 미분가능하다.

(도움정리 2.2.3) 만일 f : [a,b] \to \mathbb{R} 이 절대연속이고 거의 모든 점에서 f’ = 0 이면 f 는 상수함수이다.

(두번째 미적분학의 기본정리) (정리 2.2.4) 함수 f : [a,b] \to \mathbb{R} 에 대하여 다음은 동치이다.

함수 f 가 절대연속이다.

함수 f 가 거의 모든 점에서 미분가능하고 f’ 이 르벡적분가능하며, 등식 f(x) = f(a) + \int_{a}^{x} f’, x\in [a,b] 이 성립한다.

(정의) 임의의 단조증가함수는 절대연속함수 g와 도함수가 거의 모든 점에서 0 인 함수의 합으로 표시된다. 임의의 유계변동함수 역시 절대연속함수와 도함수가 거의 모든 점에서 0인 함수의 합으로 표시된다. 이를 르벡 분해라 한다.

(명제 2.2.5) 함수 f : [a,b] \to \mathbb{R} 가 절대연속이고 N \subset [a,b] 이 영측도집합이면 f(N) 도 영측도집합이다.

(따름정리 2.2.6) 함수 f : [a,b] \to \mathbb{R} 이 절대연속이고 E \subset [a,b]가 잴 수 있는 집합이면 f(E) 도 잴 수 있는 집합이다.

적분가능함수공간

(section 3) 적분가능함수공간

(subsection 3.1)L^{p} 공간

(정의) 잴 수 있는 집합 E \subset \mathbb{R} 과 양수 p >0 에 대하여 조건 \int_{E} |f|^{p} < \infty 를 만족하는 잴 수 있는 함수 f : E \to \mathbb{C} 의 집합을 L^{p}(E) 라 하자.

(정의) 함수 \phi : (a,b) \to \mathbb{R} (단 -\infty \le a < b \le +\infty) 이 다음 조건을 만족할 때, \phi를 볼록함수라 부른다.

a<s<t<u<b \implies \frac{\phi(t) - \phi(s)}{t-s} \le \frac{\phi(u) - \phi(t)}{u-t}

(옌센 부등식)(정리 3.1.1) 함수 f : [0,1] \to (a,b) 가 적분가능하고 \phi : (a,b) \to \mathbb{R} 이 볼록함수이면 (단 -\infty \le a < b \le +\infty) 부등식 \phi\biggl \int_{0}^{1} f \biggr \le \int_{0}^{1} (\phi \bullet f) 이 성립한다.

(정의) 두 양수 0 < p,q < \infty 가 다음 관계식 \frac{1}{p} + \frac{1}{q} 를 만족하면 이를 켤레지수라 부른다. 1, \infty 도 켤레지수라 한다.

(횔더 부등식, 민코프스키 부등식) (정리 3.1.2) 두 양수 p, q 가 켤레지수일 때, 잴 수 있는 집합 E에서 정의된 잴 수 있는 함수 f , g : E \to [0,\infty] 에 대하여 부등식이 성립한다.

\int_{e} fg \le \biggl \int_{E} f^{p}\biggr^{1/p} \biggl \int_{E} g^{p}\biggr^{1/q}

\biggl \int_{E} (f+g)^{p}\biggr^{1/p} \le \biggl \int_{E} f^{p}\biggr^{1/p} + \biggl \int_{E} g^{p}\biggr^{1/q}

(정의) 잴 수 있는 함수 f : E \to \mathbb{C} 와 p \in [1,\infty) 에 대하여 \lVert f \rVert_{p} = \biggl \int_{E} |f|^{p}\biggr^{1/p} 라 정의한다. 그러면 \lVert fg \rVert_{1} \le \lVert f \rVert_{p} \lVert g \rVert_{q} (단 1, < p,q < \infty, \frac{1}{p} + \frac{1}{q}) \lVert f + g \rVert_{p} \le \lVert f \rVert_{p}+ \lVert g \rVert_{p} (단 1 \le p < \infty) 은 횔더 부등식과 민코프스키 부등식의 또다른 형태이다. 따라서 L^{p}(E) 가 벡터공간이 된다.

(정의) 잴 수 있는 함수 f : E \to [0,\infty] 에 대하여 esssup f = \inf{\alpha \in \mathbb{R} : \mu (f^{-1}((\alpha, \infty\rbrack)) = 0 } 이라 정의하자. 위 정의에 나오는 집합이 비어있으면 esssup f = \infty 라 정의한다.

(정의) 잴 수 있는 함수 f : E \to \mathbb{C} 에 대하여 \lVert f \rVert_{\infty} = esssup|f| 라 정의한다. \lVert f \rVert_{\infty} < \infty 인 잴 수 있는 함수 f : E \to \mathbb{C} 전체의 집합을 L^{\infty} (E) 로 쓴다.

(정의) 복소벡터공간 X 에 함수 \lVert \rVert : X \to [0,\infty) 가 주어져서 다음 성질들을 만족하면 이를 노음공간이라 한다.

x \in X, \lVert x \rVert = 0 \implies x = 0

\alpha \in \mathbb{C}, x \in X \implies \lVert \alphax \rVert = |\alpha| \lVert x \rVert

x, y \in X \implies \lVert x+y \rVert \le \lVert x \rVert + \lVert y \rVert

(명제) 노음공간 L^{p} (E) 의 원소는 ‘거의 모든 점에서 함수값이 일치한다’는 동치관계에 의한 동치류에 의하여 생기는 함수 모임이다. 그러나 그 동치류에 속하는 함수 하나를 마치 L^{p} (E)의 원소인 것처럼 쓰기로 한다.

(정의) 노음공간의 수열 \langle x_{n} \rangle 이 주어져 있을 때, 임의의 양수 \epsilon > 0 에 대하여 다음 성질 m , n > N \implies \lVert x_{n} – x_{m} \rVert < \epsilon 을 만족하는 자연수 N이 존재하면 이 수열 \langle x_{n} \rangle 을 코시수열이라 한다.

(정의) 모든 코시수열이 수렴하는 노음공간을 바나하공간이라 한다.

(리쓰-피셔)(정리 3.1.3) 각 p \in [1,\infty] 에 대하여 L^{p} (E) 는 바나하공간이다.

(명제 3.1.4) 만일 L^{p}(E) (단, 1 \le p \le \infty) 의 수열 \langle f_{n} \rangle 이 f \in L^{p}(E) 로 수렴하면 (즉, \lVert f_{n} - f \rVert_{p} \to 0 이면) 거의 모든 점에서 f로 점별수렴하는 부분수열을 가진다.

(subsection 3.2) 잴 수 있는 함수와 연속함수

(도움정리 3.2.1) 임의의 유계닫힌집합 K \subset \mathbb{R} 와 열린집합 U \subset \mathbb{R} 가 K \subset U 이면, 다음을 만족하는 연속함수 f : \mathbb{R} \to [0,1] 이 존재한다.

0 \le f \le 1, f|{K} = 1, f|{\mathbb{R} \setminus U} = 0

(루진)(정리 3.2.2) 유계구간 [a,b] 위에서 정의된 잴 수 있는 함수 f : [a,b] \to \mathbb{C} 가 있을 때, 임의의 양수 \epsilon >0 에 대하여 \mu({x \in [a,b] : f(x) \neq g(x)}) < \epsilon 를 만족하는 연속함수 g : [a,b] \to \mathbb{C} 가 존재한다. 만일 f \in L^{\infty} [a,b] 이면 \lVert g\rVert _{\infty} \le \lVert f\rVert _{\infty} 가 되도록 연속함수 g 를 택할 수 있다.

(정의) 구간 [a,b] 위에서 정의된 연속함수 f : [a,b] \to \mathbb{C} 를 모두 모은 벡터공간 C[a,b]

(도움정리 3.2.3) 구간 [a,b] 에서 정의된 단순함수 s : [a,b] \to \mathbb{C} 전체의 집합 S는 L^{p}[a,b] 안에서 조밀하다. (단, 1 \le p < \infty)

(정리 3.2.4) 각 p \in [1,\infty) 에 대하여, C[a,b] 는 L^{p}[a,b] 의 조밀한 부분공간이다.

(정의) 함수 f : \mathbb{R} \to \mathbb{C} 의 받침 supp f 를 집합 {x : f(x) \neq 0} 의 닫힘으로 정의하고, 받침이 유계구간에 들어가는 경우 연속함수 전체의 벡터공간을 C_{c}(\mathbb{R}) 이라 쓴다.

(정리 3.2.5) 각 p \in [1,\infty) 에 대하여, C_{c}(\mathbb{R}) 은 L^{p}(\mathbb{R}) 의 조밀한 부분공간이다.

(정의) 집합 X의 곱집합 X \times X 에서 정의된 함수 d : X \times X \to [0,\infty) 가 다음 성질들을 만족하면 d를 거리라 하고, 거리가 주어진 집합을 거리공간이라 한다.

d(x,y) = 0 \implies x = y 이다.

d(x,y) = d(y,x) 이다.

d(x,y) \le d(x,z) + d(z,y) 이다.

(정의) 임의의 코시수열이 수렴하는 거리공간을 완비거리공간이라 한다.

거리공간의 부분집합은 다시 거리공간이 되며, 이를 부분공간이라 한다.

\tilde{X} 가 완비거리공간이고 X가 \tilde{X} 의 조밀한 부분공간이면 \tilde{X} 를 X의 완비화공간이라 한다.

(명제) 임의의 노음공간은 거리공간이다.

L^{p}(\mathbb{R}) 이 거리공간 (C_{c}(\mathbb{R}), \lVert \rVert_{p}) 의 완비화공간이다.

(정의) 실수축 위에서 정의된 연속함수 가운데 lim_{x \to \mp \infty} f(x) = 0 인 것들을 모두 모은 벡터공간을 C_{0}(\mathbb{R}) 이라 쓰자. 이 공간에 노음 \lVert f\rVert_{sup} = \sup{|f(x)| : x \in \mathbb{R}} 을 부여하면, 노음공간이 된다.

(정의) 실수축에 정의된 함수 f : \mathbb{R} \to \mathbb{C} 와 점 y \in \mathbb{R} 에 대하여 그 평행이동 f_{y} 를 f_{y}(x) = f(x-y), x \in \mathbb{R} 로 정의한다.

(정리 3.2.6) 각 함수 f \in L^{p}(\mathbb{R}) 에 대하여 (단, 1 \le p < \infty) y \mapsto f_{y} : \mathbb{R} \to L^{p}(\mathbb{R}) 은 고른연속함수이다.

(도움정리 3.2.7) 유한측도집합 E \subset \mathbb{R} 에서 정의된 잴 수 있는 복소함수열 \langle f_{n} \rangle 이 거의 모든 점에서 f : E \to \mathbb{C} 로 수렴한다고 가정하자. 그러면, 임의의 양수 \epsilon >0 과 \delta >0 에 대하여 다음 성질과 \mu(A) < \delta 를 만족하는 집합 A \subset E 와 자연수 N 을 잡을 수 있다.

n \ge N, x \in E \setminus A \implies |f(x) – f_{n}(x)| < \epsilon

(에고로프)(정리 3.2.8) 유한측도집합 E \subset \mathbb{R} 에서 정의된 잴 수 있는 복소함수열 \langle f_{n} \rangle 이 거의 모든 점에서 f : E \to \mathbb{C} 로 수렴한다고 가정하자. 그러면 임의의 양수 \epsilon >0 에 대하여 다음 성질을 만족하는 집합 A \subset E 를 잡을 수 있다.

\mu(A) < \epsilon, \langle f_{n} \rangle 이 E \setminus A 위에서 고르게 수렴한다.

(subsection 3.3) 양측도와 적분

(정의) 집합 X의 부분집합모임 \mathfrak{S}가 다음 성질을 만족할 때, 이를 \sigma-대수라 부른다.

\empty \subset \mathfrak{S }

만일 A \in \mathfrak{S} 이면 A^{c} \in \mathfrak{S} 이다.

각 n = 1,2, … 에 대하여 A_{n} \in \mathfrak{S} 이면 \bigcup_{n = 1}^{\infty} A_[n] \in \mathfrak{S} 이다.

(정의) 만일 함수 \mu : \mathfrak{S} \to [0,\infty] 가 다음 성질을 만족하고 \mu(\empty) = 0 이면, 이를 양측도 혹은 그냥 측도라 한다. 또한, 집합모임 \mathfrak{S} 에 들어가는 X의 부분집합들을 잴수있는 집합이라 부른다.

{A_{n} \in \mathfrak{S} : n = 1,2, …} 가 서로소 \implies \mu\biggl \bigsqcup_{n=1}^{\infty} A_{n} \biggr = \sum_{n = 1}^{\infty} \mu(A_{n})

(명제 3.3.1) 양측도 (X, \mathfrak{S}, \mu) 에 대하여 다음이 성립한다.

A \subset B 이면 (단 A, B \in \mathfrak{S}) \mu(A) \le \mu(B) 이다.

잴수있는 단순증가 집합열 \langle A_{n} \rangle 에 대하여 등식 \mu \biggl \bigcup_{n=1}^{\infty} A_{n} \biggr = \lim_{n \to \infty} \mu(A_{n}) 이 성립한다.

잴 수 있는 단순감소 집합열 \langle A_{n} \rangle 에 대하여 \mu(A_{1}) < \infty 이면 등식 \mu \biggl \bigcap_{n=1}^{\infty} A_{n} \biggr = \lim_{n \to \infty} \mu(A_{n}) 가 성립한다.

(정의) 다음 세 가지 성질을 만족하는 함수 \mu : \mathcal{P} (X) \to [0,\infty] 를 집합 X의 외측도라 한다.

\mu(\empty) = 0

만일 S \subseteq T 이면 \mu(S) \le \mu(T) 이다.

임의의 셀 수 있는 집합 모임 {S_{n} : n = 1,2,…} 에 대하여 \mu \biggl( \bigcup_{n = 1}^{\infty} S_{n} \biggr) \le \sum_{n=1}^{\infty} \mu(S_{n}) 이 성립한다.

(정의) 카라테오도리 조건 S \subseteq X \implies \mu(S) = \mu(S \cap E) + \mu(S \cap E^{c}) 를 만족하는 집합 E \subset X 들의 모임을 \mathfrak{S} 라 한다.

(명제 3.3.2) 집합 X에 외측도 \mu : \mathcal{P} (X) \to [0, \infty] 가 주어져 있을 때, 카라테오도리 조건을 만족하는 집합모임 \mathfrak{S} 는 \sigma – 대수가 되고, \mu(S) = 0 이면 S \in \mathfrak(S) 이다. 또한 \mu 를 \mathfrak(S) 에 제한하면 측도가 된다.

(명제 3.3.3) 집합 X 에서 정의된 함수 f : X \to \mathbb{R}^{*} 에 대하여 다음은 동치이다.

함수 f는 잴 수 있는 함수이다.

임의의 a \in \mathbb{R} 에 대하여 {x \in E : f(x) \ge a} \in \mathfrak{S} 이다.

임의의 a \in \mathbb{R} 에 대하여 {x \in E : f(x) < a} \in \mathfrak{S} 이다.

임의의 a \in \mathbb{R} 에 대하여 {x \in E : f(x) \le a} \in \mathfrak{S} 이다.

(명제 3.3.4) 집합 X에서 정의된 함수 f : X \to [-\infty, \infty] 에 대하여 다음이 성립한다.

만일 f, g가 잴 수 있고 \alpha \in \mathbb{R} 이면 f + g, fg, \alpha f 도 잴 수 있다.

각 n = 1,2, …. 에 대하여 f_{n} 이 잴 수 있으면 함수 x \mapsto \inf_{n} f_{n} (x) 와 x \mapsto \sup_{n}f_{n}(x) 도 잴 수 있다.

각 n = 1,2, …. 에 대하여 함수 f_{n} 이 잴 수 있으면 점별극한함수 \limsup_{n} f_{n} , \liminf_{n} f_{n}, \lim_{n} f_{n} 도 잴 수 있다.

함수 f : E \to \mathbb{R}^{*} 이 잴 수 있고, g : \mathbb{R} \to \mathbb{R} 이 연속이면 그 합성함수 g \bullet f 도 잴 수 있다. 여기서, f(x) = \mp \infty 인 x \in E 에 대해서는 (g \bullet f) (x) = \mp \infty 로 정의한다.

(정의) 잴 수 있는 함수 f : E \to \mathbb{R} 의 치역이 유한집합이면, 이를 단순함수라 한다.

치역이 {c_{1}, …, c_{n}} 인 단순함수 s : E \to \mathbb{R} 가 있을 때, 각 i = 1,2,…, n 에 대하여 E_{i} = { x \in E : s(x) = c_{i}} 라 두면 {E_{i}} 는 서로소이고 s 는 s = \sum_{i = 1}^{n} c_{i} \chi_{E_{i}} 같이 잴 수 있는 특성함수들의 선형결합으로 표시할 수 있다.

(정의) 정의된 단순함수가 s \ge 0 일때, 그 적분을 \int_{E} s d\mu = \sum_{i = 1}^{n} c_{i} m\muE_{i}) 라 정의한다.

(정의) 잴 수 있는 함수 f : E \to [0, +\infty] 의 적분을 \inf_{E} f = sup{\inf_{E} s : 0 \le s \le f, s 는 단순함수} 로 정의한다.

(정의) 함수 f : X \to \mathbb{R}^{*} 에 대하여 f_{+} (x) = max{f(x),0}, f_{-} (x) = - min{f(x),0}, x \in X 라 두면 f_{+}, f_{-} \ge 0 이고 f = f_{+} – f_{-} 이다.

(정의) 함수 f 의 적분을 \int_{E} f d\mu = \int_{E} f_{+} d\mu - \int_{E} f_{-} d\mu 라 정의한다. 만일 함수 f_{+} 와 함수 f_{-} 의 적분값이 모두 무한이면 f의 적분을 정의하지 않는다.

(정의) 함수 f_{+}와 f_{-} 의 적분값이 모두 유한일 때, 함수 f 를 적분가능함수라 한다.

(정의) 잴 수 있는 집합 E에서 정의된 복소함수 f : E \to \mathbb{C} 에 대해 함수 f = u + iv 의 실수부를 u, 허수부를 v라 하자. u와 v가 잴 수 있으면 f가 잴 수 있다고 한다.복소함수의 적분을 \int_{E} (u + iv) d\mu = \int_{E} u d\mu + i \int_{E} v d\mu 라 정의한다. 적분가능성은 똑같이 정의한다.

(명제) 서로소인 잴 수 있는 집합 모임 {A_{n} : n = 1,2, …} 에 대하여 A = \sqcup_{n = 1}^{\infty} A_{n} 이라 하자. 그러면 임의의 잴 수 있는 함수 f : X \to [0,\infty] 및 적분가능함수 f : X \to \mathbb{C} 에 대하여 등식 \int_{A} f d\mu = \sum_{n = 1}^{\infty} \int_{A_{n}} f d\mu 이 성립한다.

(명제) 잴 수 있는 함수 f, g : X \to [0,\infty] 혹은 적분가능함수 f, g : X \to \mathbb{C} 에 대하여 등식 \int_{X} (\alpha f + \beta g) d\mu = \alpha \int_{X} f d\mu + \beta \int_{X} g d\mu 가 성립한다. (단 \alpha, \beta 는 상수)

(명제)집합 X에서 [0, + \infty] 로 가는 잴 수 있는 함수열 \left \langle f_{n} \right \rangle 에 대하여, 부등식 \int_{X} (\liminf_{n \to \infty} f_{n}) d\mu \le \liminf_{n \to \infty} \int_{X} f_{n} d\mu 이 성립한다.

(명제) 함수열 \left \langle f_{n} \right \rangle 이 거의 모든 점에서 f로 수렴하는 경우, 만일 0 \le f_{1} \le f_{2} \le … 이거나 각 n = 1,2, … 에 대하여 |f_{n}| \le g 인 g \in L^{1}(X) 가 존재한다면, 등식 \lim_{n \to \infty} \int_{X} f_{n} d\mu = \int_{X} f d\mu 이 성립한다.

(명제) X에서 [0, +\infty] 로 가는 임의의 잴수있는 함수열 \langle f_{n} \rangle 에 대하여 등식 \int_{X} \biggl( \sum_{n = 1}^{\infty} f_{n} \biggr) d\mu = \sum_{n = 1}^{\infty}\int_{X} f_{n} d\mu 를 얻는다.

(명제) X 에서 \mathbb{C} 로 가는 L^{1}-함수열 \left \langle f_{n} \right \rangle 에 대하여 급수 \sum_{n = 1}^{\infty} \int_{E} |f_{n}| 가 수렴하면, 거의 모든 x \in E 에 대하여 급수 \sum_{n = 1}^{\infty} f_{n}(x) 가 절대수렴한다. 또한, 그 극한값을 f(x) = \sum_{n = 1}^{\infty} f_{n}(x) 라 두면 f \in L^{1}(X) 이고 그 적분값은 \int_{E} f d\mu = \sum_{n = 1}^{\infty} \int_{X} f_{n} d\mu 로 주어진다.

(명제) 집합 X 에서 정의된 잴 수 있는 함수 f 에 대하여 다음이 성립한다.

만일 f \ge 0 이고 \inf_{X} f = 0 이면 거의 모든 점에서 f = 0 이다.

만일 f \in L^{1} (X) 이고 임의의 잴 수 있는 집합 D \subset X 에 대하여 \int_{D} f = 0 이면 거의 모든 점에서 f = 0이다.

(명제 3.3.5) 집합 X의 측도 \mu(X) 가 유한값이고, F \subset \mathbb{C} 가 닫힌 집합이라 하자. 만일 f \in L^{1}(X, \mathfrak{S}, \mu) 가 다음 조건을 만족하면 거의 모든 점 x \in X 에 대하여 f(x) \in F 가 성립한다.

E \in \mathfrak{S}, \mu(E) >0 \implies \frac{1}{\mu(E)} \int_{E} f d\mu \in F

(정의) 집합 X의 멱집합 \mathcal{P} 는 \sigma -대수이다. 각 A \in \mathcal{P}(X) 에 대하여 c(A) 를 집합 A의 원소의 개수라 정의하면 이는 양측도가 되는데, 함수 f : X \to \mathbb{C} 의 적분 \int_{X} f dc 를 그냥 \sum_{x \in X} f(x) 라 쓴다. 그러나 \sum_{x \in X} f(x) 는 함수 f 가 f \ge 0 이거나, 적분가능할 때에만 그 의미가 있다.

(명제) 앞의 정의의 경우 X = \sqcup_{n=1}^{\infty} X_{n} 이면 등식 \sum_{x\inX} f(x) = \sum_{n=1}^{\infty}\sum_{x \in X_{n}} f(x) 이 성립한다.

(명제 3.3.6) 함수 f : X \to \mathbb{C} 에 대하여 등식 \sum_{x \in X} |f(x)| = \sup {\sum_{x \in F} |f(x)| : F \subset X 는 유한집합이다} 이 성립한다.

(따름정리 3.3.7) 각 자연수 m,n = 1, 2, … 에 대하여 a_{m,n} \in \mathbb{C} 이 주어져있을 때, 다음이 성립한다.

만일 a_{m,n} \ge 0 이면, \sum_{n = 1}^{\infty} \sum_{m = 1}^{\infty} a_{m,n} =\sum_{m = 1}^{\infty} \sum_{n = 1}^{\infty} a_{m,n} 이 성립한다.

다음 성질 \sum_{n} b_{n} < \infty, \sum_{m=1}^{\infty} |a_{m,n}| \le b_{n} , n = 1,2, … 을 만족하는 급수 \sum b_{n} 이 있으면, \sum_{n = 1}^{\infty} \sum_{m = 1}^{\infty} a_{m,n} =\sum_{m = 1}^{\infty} \sum_{n = 1}^{\infty} a_{m,n} 이 성립한다.

(정의) 함수 f : X \to \mathbb{C} 와 p \in [1,\infty) 에 대하여 \lVert f \rVert_{p} = \biggl( \int_{X} |f|^{p} d\mu \biggr)^{1/p} 라 정의하고, \lVert f \rVert_{\infty} 역시 이전과 같이 정의한다.

(정의) \lVert f \rVert_{p} < \infty 인 함수 f : X \to \mathbb{C} 전체의 집합을 L^{p}(X, \mathfrak{S},\mu), 줄여서 L^{p}(X,\mu) 혹은 L^{p}(\mu) 라 쓴다. 만일 E \in \mathbb{R} 이 잴 수 있는 집합이고 m 이 르벡측도라면 L^{p}(E,m) 을 그냥 L^{p}(E) 라 쓴다. L^{p}(\mu) 는 바나하공간이 된다.

(명제 3.3.8) 단순함수 s : X \to \mathbb{C} 가운데 \mu{x\in X : s(x) \neq 0} < \infty 를 만족하는 것들 전체의 집합 S는 L^{p}(\mu) 안에서 조밀하다. (단, 1 \le p < \infty)

(정의) 개수를 세는 측도에 의한 공간 L^{p}(X,\mathcal{P}, c) 는 \mathcal{l}^{p}(X) 라 쓴다. X가 자연수 집합이면 \mathcal{l}^{p} 라 쓴다.

(명제) \mathcal{l}^{\infty} 는 유계수열공간이고 \mathcal{l}^{1} 은 절대수렴하는 수열들을 모아놓은 공간이 된다. 노음공간 \mathcal{l}^{p}({1,2,…,n}) 은 \lVert x\rVert_{p} = (|x_{1}|^{p} + … + |x_{n}|^{p})^{1/p}, x = (x_{1}, x_{2}, …, x_{n}) 으로 노음이 부여된 노음공간 (\mathbb{C}^{n}, \lVert \rVert_{p} ) 이 된다.

(정의) 벡터공간 V 의 부분집합 S \subset V 가 다음 조건을 만족하면 S를 볼록집합이라 부른다.

x , y \in S, 0 \le t \le 1 \implies tx + (1-t)y \in S

(명제) 1 \le p < q \le \infty \implies \mathcal{l}^{p}(X) \subset \mathcal{l}^{q}(X)

\mu(X) < \infty, 1 \le p < q \le \infty \implies L^{q}(X,\mu) \subset L^{p}(X,\mu)

이중적분

(section 4) 이중적분

(subsection 4.1) 곱측도

(명제) 집합 X, Y 에 각각 양측도 (X, \mathfrak{S}, \mu) 와 (Y, \mathfrak(I), \nu) 가 주어져 있다. 집합 X \times Y 의 \sigma-대수는 \mathfrak(R) = {S\timesT \subset X\timesY : S \in \mathfrak{S}, T \in \mathfrak{T}}를 포함해야 한다.

(정의 1) 집합 E \subset X \times Y 와 x \in X, y \in Y 에 대하여 E_{x} = {y \in Y : (x,y) \in E}, E^{y} = {x \in X : (x,y) \in E} 라고 정의하자. 만일 각 x \in X, y \in Y 에 대하여 E_{x} \in \mathfrak{T}, E^{y} \in \mathfrak{S} 라면 두 함수 \phi : x \mapsto \nu(E_{x}) = \int_{Y} \chi_{E}(x,y) d\nu(y), \psi : y \mapsto \nu(E^{y}) = \int_{X} \chi_{E}(x,y) d\mu(y) 로 정의할 수 있다.

(정의 2) 만일 두 함수 \phi, \psi 가 각각 잴 수 있는 함수이고 등식 \int_{X} \phi d\mu = \int_{Y} \psi d\nu 이면 이 공통값을 E의 측도로 정의한다.

(정의) X와 Y가 유한측도를 가지는 부분집합들로 분할될 수 있다 가정한다. 즉, X = \bigsqcup_{n = 1}^{\infty} X_{n}, Y=\bigsqcup_{m = 1}^{\infty} Y_{m} 이라 가정한다. (단, 각 m, n = 1,2,…에 대하여 \mu(X_{n}), \nu(Y_{m}) < \infty) 두 집합 모임 \mathfrak{B} = {E \in \mathfrak{S} \times \mathfrak{T} : (정의 1) 에서 정의한 두 함수 \phi, \psi 가 잴 수 있고 (정의 2)의 등식이 성립한다}

\mathfrak{C} = {E \in \mathfrak{S} \times \mathfrak{T} : E \cap (X_{n} \times Y_[m]) \in \mathfrak{B} (m,n = 1,2,…)}

(도움정리 4.1.1) 집합모임 \mathfrak{B}와 \mathfrak{C} 에 대하여 다음이 성립한다.

\mathfrak{B} 는 단조증가하는 집합열의 합집합에 대하여 닫혀있다.

\mathfrak{B} 는 서로소인 집합열의 합집합에 대하여 닫혀있다.

\mathfrak{C} 는 단조증가하는 집합열의 합집합과 단조감소 집합열의 교집합에 대하여 닫혀있다.

만일 E_{1} , …, E_{n} \in \mathfrak{R} 이 서로소이면 \bigsqcup_{n = 1}^{N} E_{n} \in \mathfrak{C}이다.

(정의) 전체집합 X가 유한측도를 가지는 부분집합열의 합집합으로 표시되면 , (X ,\mu) 를 \sigma -유한측도 라 한다.

(정리 4.1.2) 양측도 (X, \mathfrak{S}, \mu) 와 (Y, \mathfrak{T},\nu) 가 주어져 있을 때, 잴수있는 상자들을 포함하는 최소의 \sigma-대수를 \mathfrak{S} \times \mathfrak{T} 라 쓰자. 만일 E \in \mathfrak{S} 이면, 임의의 x \in X 와 y \in Y 에 대하여 E_{x} \in \mathfrak{T} 및 E^{y} \in \mathfrak{S 가 성립한다. 만일 (X, \mathfrak{S}, \mu) 와 (Y, \mathfrak{T},\nu) 가 각각 \sigma-유한이면 (정의 1) 의 \phi,\psi 가 각각 잴 수 있는 함수이고 (정의 2)의 등식이 성립한다.

(정의) (정의 2)의 등식의 공통값을 (\mu \times \nu)(E) 라 쓰면, (X \times Y, \mathfrak{S} \times \mathfrak{T} , \mu \times \nu) 를 \sigma -유한측도 (X, \mathfrak{S}, \mu) 와 (Y, \mathfrak{T},\nu)의 곱측도라 부른다.

(정의) 르벡측도 (\mathbb{R}, \mathfrak{M}, m) 에 대해 좌표공간 \mathbb(R)^{n} 의 열린 집합들을 포함하는 최소의 \sigma-대수를 \mathfrak{B}{n} 이라 쓰면, \mathfrak{B}{2} \subset \mathfrak{M} \times \mathfrak{M} 이 된다. 집합모임 \mathfrak{B}_{n} 에 들어가는 \mathbb{R}^{n} 의 부분집합을 보렐집합이라고 한다.

닫힌집합, G_{\delta} 집합, F_{\delta}-집합 등은 보렐집합의 예이다.

(정의) 측도 (X,\mathfrak{S}, \mu) 가 성질 E \subset F , F \in \mathfrak{S} , \mu(F) = 0 \implies E \in \mathfrak{S} 를 만족하면 이를 완비측도라 한다.

(명제 4.1.3) 양측도 (X, \mathfrak{S}, \mu) 에 대하여 집합모임 \mathfrak{S}^{} = {E \subset X : A \subset E \subset B 및 \mu(B\setminus A) = 0 을 만족하는 A,B \in \mathfrak{S} 가 존재한다.} 을 생각하고, 각 E \in \mathfrak{S}^{} 에 대하여 \mu^{}(E) = \mu(A) 라 정의하자. 그러면 (X, \mathfrak(S)^{}, \mu^{*}) 는 완비측도가 된다.

(정의) (\mathfrak{M} \times \mathfrak{M})^{} = \mathfrak{B}_{2}^{} 이다. 이 \sigma -대수를 \mathfrak{M}{2} 라 두고 이 위에서 정의된 측도 (m \times m) 을 m{2} 라 쓰면 완비측도 (\mathbb{R}^{2}, \mathfrak{M}{2}, m{2}) 를 얻는데, 이를 \mathbb{R}^{2} 의 르벡측도라 한다.

(subsection 4.2) 푸비니 정리

(정의) 지금부터 집합 X, Y 에 양측도 (X, \mathfrak{S}, \mu) 와 (Y, \mathfrak{T}, \nu) 가 주어져있다고 가정하고 X \times Y 위에 정의된 임의의 함수 f 에 대하여 f_{x} : y \mapsto f(x,y), f^{y} : x \mapsto f(x,y) 라 정의한다.

(명제 4.2.1) 함수 f : X \times Y \to \mathbb{R}^{*} 이 \mathfrak{S} \times \mathfrak{T} 에 대하여 잴 수 있는 함수라 하자. 그러면, 임의의 x \in X 와 y \in Y 에 대하여 f_{x} 와 f^{y} 는 각각 \mathfrak{T} 와 \mathfrak{S} 에 대하여 잴 수 있는 함수이다.

(토넬리)(정리 4.2.2) 양측도 (X, \mathfrak{S}, \mu) 와 (Y, \mathfrak{T}, \nu) 가 각각 \sigma-유한이라고 하자. 그러면 임의의 잴 수 있는 함수 f : X \times Y \to [0,\infty] 에 대하여 다음이 성립한다.

함수 x \mapsto \int_{Y}f_{x}d\nu 와 y \mapsto \int_{X}f^{y}d\mu 가 각각 잴 수 있는 함수이다.

등식 \int_{X}[\int_{Y}f_{x}d\nu]d\mu = \int_{X\timesY}fd(\mu\times\nu) = \int_{y} [\int_{X} f^{y}d\mu] d\nu

(푸비니)(정리 4.2.3) 양측도 (X, \mathfrak{S}, \mu) 와 (Y, \mathfrak{T}, \nu)가 각각 \sigma -유한이라고 하자. 잴 수 있는 함수 f:X \times Y \to \mathbb{C} 가 \int_{X} [\int_{Y} |f(x,y)| d\nu(y) ] d\mu(x) < \infty 를 만족하면 다음이 성립한다.

f \in L^{1}(\mu \times \nu) 이다. 또한, 거의 모든 x \in X 에 대하여 f_{x} \in L^{1}(\nu) 이고, 거의 모든 y \in Y 에 대하여 f^{y} \in L^{1}(\mu) 이다.

거의 모든 점에서 정의된 두 함수 x \mapsto \int_{Y} f_{x}d\nu 와 y \mapsto \int_{x}f^{y} d\mu 가 각각 적분가능함수이다.

등식 \int_{X}[\int_{Y} f_{x}d\nu] d\mu = \int_{X \times Y}fd(\mu \times \nu) = \int_{Y}[\int_{X} f^{y}d\mu] d\nu 이 성립한다.

(정의) 좌표공간 \mathbb{R}^{n} 위에 정의된 함수가 \sigma-대수 \mathfrak{B} 에 대하여 잴 수 있으면 이를 보렐함수라 부른다. 그냥 잴 수 있는 함수라 하면 르벡측도 (\mathfrak{M}{n}, m{n}) 에 대하여 잴 수 있음을 뜻한다.

(명제 4.2.4) 양측도 (X, \mathfrak{S} , \mu) 에 대하여 잴 수 있는 함수인 f : X \to \mathbb{R} 과 보렐함수 g : \mathbb{R} \to \mathbb{R} 에 대하여 그 합성함수 g \bullet f : X \to \mathbb{R} 은 잴 수 있는 함수이다.

(명제 4.2.5) 좌표공간 \mathbb{R}^{n} 에서 정의된 잴 수 있는 함수 f 가 주어지면, 거의 모든 점에서 f = g 인 보렐함수 g 가 존재한다.

(토넬리-푸비니)(정리 4.2.6) 좌표평면 \mathbb{R}^{2} 에서 정의된 잴 수 있는 함수 f 에 대하여 다음이 성립한다.

거의 모든 점 x 와 y 에 대하여, f_{x} 및 f^{y} 는 잴 수 있는 함수이다.

만일 0 \le f \le \infty 이면 등식 \int_{\mathbb{R}}\int_{\mathbb{R}}f(x,y) dxdy = \int_{\mathbb{R}\times \mathbb{R}} f dm_{2} = \int_{\mathbb{R}}\int_{\mathbb{R}}f(x,y) dydx 이 성립한다.

만일 f : \mathbb{R}^{2} \to \mathbb{C} 이고 \int_{\mathbb{R}}\int_{\mathbb{R}} |f(x,y)| dxdy < \infty 이면 f \in L^{1}(m_{2}) 이다. 또한, 거의 모든 x, y 에 대하여 f_{x}, f^{y} \in L^{1}(m_{1}) 이고 등식 \int_{\mathbb{R}}\int_{\mathbb{R}}f(x,y) dxdy = \int_{\mathbb{R}\times \mathbb{R}} f dm_{2} = \int_{\mathbb{R}}\int_{\mathbb{R}}f(x,y) dydx 이 성립한다.

(subsection 4.3) 적분가능함수의 곱하기

(정의) 적분가능함수 f,g \in L^{1}(\mathbb{R}) 에 대하여 x \mapsto \int_{\mathbb{R} f(y)g(x-y) dy 로 정의된 새로운 L^{1}- 함수를 f*g 라 쓰고, 이를 f 와 g의 곱하기 또는 컨볼류션이라 부른다. \lVert f * g\rVert_{1} \le \lvert f\rVert_{1} \lVert g \rVert_{1} , f,g \in L^{1}(\mathbb{R}) 이 된다.

(명제 4.3.1) 각 f,g,h \in L^{1}(\mathbb{R}) 와 상수 \alpha, \beta \in \mathbb{C} 에 대하여 다음이 성립한다.

거의 모든 점 x 에 대하여 |(fg)h|(x) = |f(gh)|(x) 가 성립한다.

각 x에 대하여 (fg)(x) = (gf)(x) 이다.

거의 모든 점 x에 대하여 |(\alpha f + \beta g)h|(x) = |\alpha(fh) + \beta(g*h)|(x) 가 성립한다.

(정의) 일반적으로, 바나하공간 X에 연산 (x,y) \mapsto xy 가 정의되어 결합법칙 및 분배법칙과 부등식 \lVert xy\rVert \le \lvert x\rVert \lVert y \rVert , x,y \in X 를 만족하면, 이를 바나하대수라 한다. L^{1}(\mathbb{R}) 은 바나하대수이다.

(명제 4.3.2) 만일 함수 g가 미분가능하고 f,g,g’ \in L^{1}(\mathbb{R}) 이면, fg 가 미분가능하고 (fg)’ = f*g’ 이 성립한다.

(명제 4.3.3) 두 양수 p,q \in (1,\infty)가 켤레지수라 하자. 그러면 임의의 f\in L^{p}(\mathbb{R}) 과 g \in L^{q}(\mathbb{R}) 에 대하여 f * g \in C_{0}(\mathbb{R}) 이다.

푸리에 변환

(section 5) 푸리에 변환

(subsection 5.1) 푸리에 급수

(정의) 실수군 \mathbb{R} 을 부분군 {2\pi n : n = 0, \mp 1, \mp 2,…} 으로 나눈 몫을 \mathbb{T} 라 두고, 실수축의 거리를 그대로 쓴다. \int_{\mathbb{T}} f = \frac{1}{2\pi} \int_{-\pi}^{\pi} f(x)dx 라 정의한다.

(정의) 각 f,g : \mathbb{T} \to \mathbb{C} 와 t \in \mathbb{T} 에 대하여 (f*g)(t) = \int_{\mathbb{T}} f(s)g(t-s) ds 라 정의한다. 그러면 L^{1}(\mathbb{T}) 는 바나하대수가 된다.

(정의) 각 f \in L^{1}(\mathbb{T}) 와 정수 n = 0, \mp 1, \mp 2, … 에 대하여 \hat{f}(n) = \int_{\mathbb{T}} f(t)e^{-int}dt 라 정의하면 |\hat{f}(n)| \le \lVert f \rVert_{1} 이다.

(정의) 각 n = 0, \mp 1, \mp 2, …., 에 대하여 u_{n}(t) = e^{int} , t\in \mathbb{T} 라 두면 관계식 u_{n} * f = \hat{f}(n)u_{n} , n \in \mathbb{Z}, f \in L^{1}(\mathbb{T}) 이 바로 확인되는데, 특히 (u_{n} * f)(0) = \hat{f}(n) 이다. 또한 다음이 성립한다.

\hat{u_{n}}(m) = \begin{cases}1&m = n \ 0, m \neq n \end{cases}

(정의) 무한급수 \sum_{n = -\infty}^{\infty} \hat{f}(n) u_{n} 을 f \in L^{1}(\mathbb{T}) 의 푸리에급수라 부르고, 그 계수 {\hat{f}(n) : n \in \mathbb{Z}} 를 f의 푸리에계수라 부른다.

(명제 5.1.1) 각 f, g \in L^{1}(\mathbb{T}) 및 상수 a,b \in \mathbb{C} 와 n \in \mathbb{Z} 에 대하여 등식들 \hat{af + bg}(n) = a\hat{f}(n) + b\hat{g}(n), \hat{f * g} (n) = \hat{f}(n) \hat{g}(n) 이 성립한다. 또한, 각 t \in \mathbb{T} 에 대하여 다음 등식 \hat{f_{t}}(n) = \hat{f}(n) u_{n}(-t), n \in \mathbb{Z} 이 성립한다.

(명제 5.1.2) 만일 f 가 주기 2\pi 인 미분가능함수고 f’ \in L^{1}(\mathbb{T} 이면 등식 \hat{f’}(n) = in \hat{f}(n), n \in \mathbb{Z} 이 성립한다.

(정리 5.1.3) 각 p \in [1,\infty) 에 대하여, 연속함수공간 C(\mathbb{T}) 는 L^{p}(\mathbb{T})의 조밀한 부분공간이다.

(정의) 집합 \mathbb{T} 에서 정의된 함수 f 와 t \in \mathbb{T} 에 대하여, 그 평행이동을 f_{t}(s) = f(s-t), s \in \mathbb{T} 라 정의한다.

(정리 5.1.4) 각 함수 f \in L^{p}(\mathbb{T}) 에 대하여 (단, 1 \le p < \infty) 함수 t \mapsto f_{t} : \mathbb{T} \to L^{p}(\mathbb{T}) 는 고른연속함수이다.

(정리 5.1.5) 함수열 \langle h_{n} \rangle 이 다음 성질 h_{n} \ge 0, \int_{\mathbb{T}} h_{n} = 1, lim_{n \to \infty} \int_{\mathbb{T}\setminus[-\delta,\delta]} h_{n} = 0 (0 < \delta < \pi) 를 만족하면, 임의의 f \in L^{1}(\mathbb{T}) 에 대하여 lim_{n \to \infty} \lVert f – h_{n} * f \rVert_{1} = 0 이 성립한다.

(정리 5.1.6) 함수열 \langle h_{n} \rangle 이 성질 h_{n} \ge 0, \int_{\mathbb{T}} h_{n} = 1, lim_{n \to \infty} \int_{\mathbb{T}\setminus[-\delta,\delta]} h_{n} = 0 (0 < \delta < \pi) 을 만족하면, 임의의 f \in C(\mathbb{T}) 에 대하여 lim_{n \to \infty} lim_{n \to \infty} \lVert f – h_{n} * f \rVert_{\infty} = 0 이 성립한다.

(정의) 각 n = 1, 2, … 에 대하여 K_{n} = \sum_{k = -n}^{n} (1-\frac{|k|}{n+1}) u_{k} 이라 정의했을 때, \langle K_{n} \rangle 을 페제르 핵이라 한다. 이는 성질 h_{n} \ge 0, \int_{\mathbb{T}} h_{n} = 1, lim_{n \to \infty} \int_{\mathbb{T}\setminus[-\delta,\delta]} h_{n} = 0 (0 < \delta < \pi) 을 모두 만족한다.

(정의) {u_{n} : n \in \mathbb{Z} }의 유한선형결합으로 표시되는 함수를 삼각다항식이라 부르고, 삼각다항식의 벡터공간을 T(\mathbb{T}) 라 쓴다. 삼각다항식 P = \sum_{n = -N}^{N} a_{n}u_{n} 의 푸리에계수 \hat{P} (n) = a_{n} 으로 주어진다.

(정리 5.1.7) 함수공간 T_{\mathbb{T}} 는 L^{1}(\mathbb{T})의 조밀한 부분공간이다.

(정리 5.1.8) 만일 f \in L^{1}(\mathbb{T}) 이고 \hat{f} = 0 이면 거의 모든 점에서 f = 0 이다.

(따름정리 5.1.9) 만일 f \in L^{1}(\mathbb{T}) 이고 \hat{f} \in \mathcal{l}^{1}(\mathbb{Z}) 이면, 거의 모든 점 t \in \mathbb{T} 에서 등식 f(t) = \sum_{n \in \mathbb{Z}} \hat{f}(n) e^{int} 이 성립한다.

(리만-르벡) (정리 5.1.10) 각 f \in L^{1}(\mathbb{T}) 에 대하여 lim_{n \to \mp \infty} \hat{f} (n) = 0 이 성립한다.

(정의) 정수집합에서 정의된 함수 a : n \mapsto a(n) 가운데 lim_{n \to \mp \infty} a(n) = 0 인 것 전체의 벡터공간을 c_{0}(\mathbb{Z}) 라 쓰고, \mathcal{l}^{\infty}(\mathbb{Z}) 의 노음을 그대로 사용한다.

(정리 5.1.11) 변환 f \mapsto \hat{f} 는 바나하대수 L^{1}(\mathbb{T}) 에서 c_{0}(\mathbb{Z}) 로 가는 노음감소 일대일 선형사상이다.

(정의) 두 함수 a, b \in c_{0}(\mathbb{Z}) 에 대하여 그 곱 a,b 를 (ab)(n) = a(n)b(n) , n \in \mathbb{Z} 라 정의하면 c_{0}(\mathbb{Z}) 는 바나하대수가 된다. 따라서 (정리 5.1.11)의 변환은 바나하대수 사이의 정의된 선형사상으로 곱하기를 보존한다. 이를 준동형이라 한다.

(따름정리 5.1.12) 함수공간 T(\mathbb{T}) 는 C(\mathbb{T}) 의 조밀한 공간이다.

(따름정리 5.1.13) 함수공간 T(\mathbb{T}) 는 L^{p}(\mathbb{T}) (단, 1 \le p < \infty)의 조밀한 공간이다.

(바이에르쉬트라스) (정리 5.1.14) 구간 [0,1] 위에서 다항식으로 정의된 함수들의 공간은 C[0,1] 에서 조밀하다.

(subsection 5.2) 푸리에적분

(정의) 임의의 f \in L^{1}(\mathbb{R}) 과 \alpha \in \mathbb{R} 에 대하여 푸리에적분 \hat{f} (\alpha) 를 \hat{f}(\alpha) = \int_{\mathbb{R}} f(x)e^{-i \alpha x} dx 과 같이 정의한다. 그러면 |\hat{f} (\alpha )| \le \lVert f \rVert_{1} , \alpha \in \mathbb{R} 이 성립한다.

(정의) 임의의 f \in L^{1}(\mathbb{R}) 에 대하여 \hat{f} 는 유계연속함수가 되는데, \hat{f} 를 f 의 푸리에변환이라 한다. 각 \alpha \in \mathbb{R} 에 대하여 u_{\alpha}(x) = e^{i \alpha x} , x \in \mathbb{R} 로 나타내면 등식 u_{\alpha} * f = \hat{f} (\alpha) u_{\alpha}, \alpha \in \mathbb{R} 이 성립한다.

(명제 5.2.1) 각 f, g \in L^{1}(\mathbb{R}) 및 상수 a,b \in \mathbb{C} 와 n \in \mathbb{Z} 에 대하여 등식들 \hat{af + bg}(n) = a\hat{f}(n) + b\hat{g}(n), \hat{f * g} (n) = \hat{f}(n) \hat{g}(n) , \alpha \in \mathbb{R} 이 성립한다. 또한, 각 \alpha \in \mathbb{R} 에 대하여 다음 등식 \hat{f u_{\alpha}} = (\hat{f}){\alpha} , \hat{f{x}}(\alpha) = \hat{f}(\alpha) u_{\alpha}(-x), x \in \mathbb{R} 이 성립한다.

(명제 5.2.2) 함수 f \in L^{1}(\mathbb{R}) 에 대하여 다음이 성립한다.

각 x \in \mathbb{R} 에 대하여 g(x) = -ixf(x) 라 정의하였을 때, 만일 g \in L^{1}(\mathbb{R}) 이면 \hat{f} 가 미분가능하고 등식 \hat{f}’ (\alpha) = \hat{g}(\alpha) , \alpha \in \mathbb{R} 이 성립한다.

만일 f 가 미분가능하고 f’ \in L^{1}(\mathbb{R}) 이면 등식 \hat{f’} (\alpha) = i \alpha \hat{f} ( \alpha ), \alpha \in \mathbb{R} 이 성립한다.

(정리 5.2.3) 각 f \in L^{1}(\mathbb{R}) 에 대하여 \hat{f} \in C_{0}(\mathbb{R}) 이다.

(정리 5.2.4) 함수모임 { h_{\lambda} : \lambda \in (0, \infty) } 이 다음 성질 h_{\lambda} \ge 0, \int_{\mathbb{R}} h_{\lambda} = 1, lim_{\lambda \to \infty} \int_{\mathbb{R}\setminus[-\delta,\delta]} h_{\lambda} = 0 (0 < \delta ) 를 만족하면 다음이 성립한다.

임의의 f \in L^{1} (\mathbb{R}) 에 대하여 lim_{\lambda \to \infty} \lVert f – h_{\lambda} * f \rVert_{1} = 0 이 성립한다.

임의의 유계 고른연속함수 f : \mathbb{R} \to \mathbb{C} 에 대하여 lim_{\lambda \to \infty} \lVert f – h_{\lambda} * f \rVert_{\infty} = 0 이 성립한다.

(명제) 함수 h : \mathbb{R} \to \mathbb{C} 가 다음 두 조건 h \ge 0 , \int_{\mathbb{R}} h = 1 을 만족할 때, h_{\lambda} (x) = \lambda h(\lambda x), x \in \mathbb{R}, \lambda > 0 이라 두면 {h_{\lambda} : \lambda > 0} 가 성질들 h_{\lambda} \ge 0, \int_{\mathbb{R}} h_{\lambda} = 1, lim_{\lambda \to \infty} \int_{\mathbb{R}\setminus[-\delta,\delta]} h_{\lambda} = 0 (0 < \delta ) 를 만족한다. 두 조건을 만족시키는 함수 h가 h(x) = \frac{1}{2\pi} \int_{\mathbb{R}} p(\alpha)u_{\alpha}(x) d\alpha 일 때 매우 유용하게 쓰인다.

(정의) H(x) = \frac{1}{2\pi} \int_{\mathbb{R}} e^{-|\alpha|} e^{i \alpha x} d\alpha = \frac{1}{\pi} \frac{1}{1+x^{2}} 라 두면 위 명제의 조건을 충족하며, H_{\lambda} = \frac{1}{2\pi} \int_{\mathbb{R}} e^{-|\alpha/\lambda|} e^{i \alpha x} d\alpha = \frac{1}{2 \pi} \frac {2\lambda}{1+\lambda^{2} x^{2}} 이 되는데, {H_{\lambda} : \lambda > 0 } 을 뽀아송 핵이라 부른다.

(정의) 실수 \mathbb{R} 에서 정의된 함수 f \in C_{c}(\mathbb{R}) 가운데 n 번 미분가능하고 n계 도함수 f^(n) 이 연속인 함수들의 모임을 C_{c}^{n}(\mathbb{R}) 이라 쓰고, C_{c}^{\infty} (\mathbb{R}) = \bigcup_{n=1}^{\infty} C_{c}^{n}(\mathbb{R}) 이라 두면 이는 벡터공간이 된다.

(명제 5.2.5) 함수공간 C_{c}^{\infty} (\mathbb{R}) 는 노음공간 L^{1}(\mathbb{R}) 이나 C_{0} (\mathbb{R}) 에서 조밀한 부분공간이다.

(역변환공식) (정리 5.2.6) 만일 f \in L^{1}(\mathbb{R}) 이고 \hat{f} \in L^{1}(\mathbb{R}) 이면, 등식 f(x) = \frac{1}{2 \pi} \int_{\mathbb{R}} \hat{f}(\alpha) e^{i \alpha x} d\alpha 이 거의 모든 점 x 에서 성립한다.

f가 연속함수이면 등식은 모든 x \in \mathbb{R} 에 대하여 성립한다.

(정리 5.2.7) 만일 f \in L^{1}{\mathbb{R}} 이고 \hat{f} = 0 이면 거의 모든 점에서 f = 0 이다.

(정의) 각 \lambda > 0 에 대하여 K_{\lambda} (x) = \frac{1}{2 \pi} \int_{\mathbb{R}} \max{1- \frac{|\alpha|}{\lambda}, 0} u_{\alpha}(x) d\alpha = \frac{1}{2 \pi} \int_{\mathbb{R}} \biggl( 1- \frac{|\alpha|}{\lambda} \biggr) e^{i \alpha x} d\alpha 라 두면 함수모임 {K_{\lambda} : \lambda > 0} 을 페제르 핵이라 부른다.

(정의) G(x) = \frac{1}{2 \pi} \int_{\mathbb{R}} e^{-\alpha^{2}/2}u_{\alpha}(x) d\alpha 라 두자. 이로부터 얻어지는 함수모임 {G_{\lambda} : \lambda > 0} 을 가우스 핵이라 부른다.

(명제) g 와 \hat{g} 가 L^{1}-연속함수일 때 등식 g(x) = \frac{1}{2\pi} \hat{\hat{g}}(-x) , x \in \mathbb{R} 이 성립함을 말한다.

(명제) \hat{K_{\lambda}} (\alpha) = \max{1- \frac{|\alpha|}{\lambda}, 0}, \hat{P_{\lambda}} (\alpha) = e^{-|\alpha/\lambda|}, \hat{G_{\lambda}} (\alpha) = e^{-\alpha^{2}/2\lambda^{2}}

(정리 5.2.8) 푸리에 변환 f \mapsto \hat{f} 는 바나하공간 L^{1}(\mathbb{R}) 에서 C_{0} (\mathbb{R}) 로 가는 일대일 노음감소 중동형이고 그 치역이 조밀하다. 단 C_{0}(\mathbb{R})의 곱하기는 점별곱하기 (fg)(x) = f(x)g(x) 로 주어진다.

(정의) 르벡적분가능함수 f \in L^{1}(\mathbb{R}) 이 주어지면, 급수 F(t) = \sum_{n = -\infty}^{\infty} f(t-2n \pi) 가 거의 모든 점 t \in \mathbb{T} 에서 절대수렴하고 F \in L^{\mathbb{T}} 가 된다. F의 푸리에 급수가 t \in \mathbb{T} 에서 점별수렴한다면 등식 \sum_{n = -\infty}^{\infty} f(t – 2n\pi) = \frac{1}{2\pi} \sum_{n = -\infty}^{\infty} \hat{f}(n) e^{int} 를 얻는데, 이는 푸리에급수와 푸리에 적분의 관계를 보여주며 를 뽀아송 등식이라 한다.

노음공간

(section 6) 노음공간

(subsection 6.1) 쌍대공간

(정의) 두 노음공간 사이에 정의된 선형사상 T : X \to Y 에 대하여 \lVert T\rVert = sup{\lVert T(x) \rVert : x \in X, \lVert x \rVert \le 1} 이라 정의하자. 만일 \lVert T\rVert < \infty 이면 T 를 유계선형사상이라 부르고 \lVert T\rVert 를 T의 노음이라 한다.

선형사상 T : X \to Y 의 노음이란 X의 단위공 {x \in X : \lVert x \rVert \le 1} 의 치역이 Y에서 얼마나 늘어나는가 재는 것이다.

(명제 6.1.1) 노음공간 사이의 선형사상 T : X \to Y 에 대하여 다음은 동치이다.

T가 유계사상이다

T가 연속이다.

T가 한 점 x_{0} \in X 에서 연속이다.

(명제 6.1.2.2) 임의의 노음공간 X 에서 Y 로 가는 유계선형사상 전체의 집합 B(X,Y) 는 노음공간이 된다. 만일 Y가 바나하공간이라면 B(X,Y) 도 바나하공간이다.

(정의) 그 치역이 스칼라 \mathbb{C} 인 선형사상을 선형범함수라 하고, X에서 정의된 유계선형범함수 전체의 노음공간을 X^{} 라 표시한다. 즉, X = B(X,\mathbb{C}) 이다. X* 은 항상 바나하공간이 되는데, X^{*}를 X의 쌍대공간이라 한다.

(정의) 양측도 (X, \mu) 가 주어지면 임의의 q \in [1,\infty]에 대하여 g \mapsto \phi_{g} : L^{q} (\mu) \to L^{p}(\mu)^{*} 가 선형사상임이 바로 확인되는데, 이 사상이 노음을 보존한다는 것을 보였다. (단,p = 1, q = \infty 인 경우 (X, \mu) 가 \sigma-유한이라 가정한다) 이렇게 노음을 보존하는 선형사상을 등거리사상이라 부른다.

등거리사상은 단사사상이다.

(정리 6.1.3) 만일 1 \le p < \infty 이면 g \mapsto \phi_{g} : \mathcal{l}^{q} \to (\mathcal{l}^{p})^{*} 은 전단사 등거리사상이다. (단, p,q 는 켤레지수)

(한-바나하)(정리 6.1.4) 노음공간 X와 그 부분공간 Y가 주어져있다. 그러면 임의의 \psi \in Y^{} 에 대하여 다음 성질 \phi |_{Y} = \psi, \lVert \phi \rVert = \lVert \psi \rVert 을 만족하는 \phi \in X^{} 가 존재한다.

(따름정리 6.1.5) 노음공간 X의 점 x 가 x \neq 0 이면 \phi(x) = \lVert x \rVert 이고 \lVert \phi \rVert 인 \phi \in X^{*} 가 존재한다.

따라서, 노음공간 X는 항상 X^{**} 의 부분공간이다.

(정의) 만일 x \mapsto T_{x} : X \to X^{**} 가 전사사상이면 X를 반사공간이라 부른다.

그 예로 바나하공간 \mathcal{l}^{p} (1 < p < \infty) 는 반사공간이다.

부분집합 Y 의 닫힘은 \bar{Y}로 표시한다.

(따름정리 6.1.6) 노음공간 X의 부분공간 Y와 한 점 x \in X 에 대하여 다음은 동치이다.

x \in \bar{Y} 이다.

\phi \in X^{*} 이고 \phi |_{Y} = 0 이면 \phi (x) = 0 이다.

(subsection 6.2) 바나하공간의 선형사상

(정의)거리공간 (X,d) 의 점 x \in X 와 양수 r > 0 에 대하여 x 의 근방을 N_{X} (x,r) = {y \in X : d(x,y) < r} 로 정의하고, N(x,r)로 쓴다. X가 노음공간일 때 0 의 근방 N_{X}(0,r) 은 X_{r} 이라 쓴다.

(베르) (정리 6.2.1) 완비거리공간 X의 닫힌집합열 \langle C_{n} \rangle 의 합집합이 전체공간이 되면, 그 중 한 닫힌 집합은 열린집합을 품는다.

(따름정리 6.2.2) 완비거리공간의 조밀 열린집합열의 교집합은 비어있지 않다.

(열린사상정리)(정리 6.2.3) 바나하공간 사이에 정의된 유계선형사상 T : X \to Y 가 전사사상이라 하자. 그러면 T(X_{1}) \supset Y_{delta} 인 양수 \delta >0 가 존재한다.

(따름정리 6.2.4) 바나하공간 사이에 정의된 전사 유계선형사상 T : X \to Y 는 항상 열린사상이다.

(따름정리 6.2.5) 바나하공간 사이에 정의된 유계선형사상 T : X \to Y 가 전단사사상이면 그 역사상 T^{-1} : Y \to X 도 유계사상이다.

(정의) 각 n = 1,2, … 에 대하여 D_{n}(T) = \sum_{k = -n}^{n} u_{k}(t) = \frac{sin(n+\frac{1}{2}) t}{sin \frac{1}{2} t} , t \in \mathbb{T} 라 정의하자. 함수열 \langle D_{n} \rangle 은 디리끌렛 핵이라 불린다.

(명제) 주기함수의 푸리에 변환 L^{1}(\mathbb{T}) \to c_{0}(\mathbb{Z}) 이 전사사상이 아니다.

(닫힌그래프정리)(정리 6.2.6) 바나하공간 사이의 선형사상 T : X \to Y 가 조건 x_{n} \in X, lim_{n \to \infty} x_{n} = x, lim_{n \to \infty} T(x_{n}) = y \implies T(x) = y 를 만족하면 T는 유계선형사상이다.

(고른유계원칙) (바나하-쉬타인하우스 정리) (정리 6.2.7) 바나하공간 X 에서 노음공간 Y 로 가는 유계선형사상들의 모임 {T_{\alpha} : \alpha \in A} 가 주어져있다. 만일 각 x \in X 에 대하여 M(x) = \sup{\lVert T_{\alpha} (x) \rVert: \alpha \in A} < \infty 이면 \sup{\lVert T_{\alpha} \rVert: \alpha \in A} < \infty 이다.

(따름정리 6.2.8) 바나하공간 X에서 노음공간 Y로 가는 유계선형사상열 \langle T_{n} \rangle 이 있는데, 각 x \in X 에 대하여 Y 의 수열 \langle T_{n} (x) \rangle 이 수렴한다고 가정하자. 이 때, T(x) = \lim_{n \to \infty} T_{n} (x) 라 두면 T : X \to Y 는 유계선형사상이다.

(명제) 푸리에 급수가 점별수렴하지 않는 연속함수가 존재한다.

(subsection 6.3) 힐버트공간

복소벡터공간 X가 있을 때, 함수 (x,y) \mapsto \langle x,y \rangle : X \times X \to \mathbb{C} 가 다음 성질들을 만족하면 이를 내적이라 하고, 내적이 정의되어있는 벡터공간을 내적공간이라 한다.

각 y \in Y 에 대하여 x \mapsto \langle x,y \rangle : X \to \mathbb{C} 가 선형범함수이다.

각 x, y \in X 에 대하여 \langle x,y \rangle = \bar{\langle y,x \rangle}

각 x \in X 에 대하여 \langle x,x \rangle \ge 0 이다.

x = 0 \iff \langle x,x \rangle = 0 이다.

(정의) 부등식 | \langle x,y \rangle |^{2} \le \langle x,x \rangle \langle x,y \rangle , x, y \in X 를 얻는데, 이를 쉬바르츠 부등식이라 한다.

(명제) 각 x \in X 에 대하여 \lVert x \rVert = \langle x,x \rangle^{1/2} 라 정의하면 x \mapsto \lVert x \rVert 가 노음이 된다. 따라서 임의의 내적공간은 노음공간이 된다.

(정의) 완비내적공간을 힐버트 공간이라 한다.

(정의) 내적공간의 노음은 등식 \lVert x+y \rVert^{2} + \lVert x-y \rVert^{2} = 2 \lVert x \rVert^{2} + 2 \lVert y \rVert^{2} 을 만족하는데, 이를 나란히꼴 등식이라 부른다.

(명제) 임의의 y \in H 에 대하여 \phi_{y} : X \mapsto \langle x,y \rangle , x \in H 는 유계선형범함수이고, \lVert \phi_{y} \rVert = \lVert y \rVert 이다. 따라서 y \mapsto \phi_{y} : X \to X^{*} 는 등거리 단사사상이다. 선형사상은 아니다. (정리 6.3.3) 에 의해 전사사상이다.

(정의) 각 \alpha \in \mathbb{C} 에 대하여 \phi_{\alpha y} = \bar{\alpha} \phi_{y} 가 되는 사상을 켤레선형사상이라 부른다.

(정리 6.3.1) 힐버트 공간 H의 닫힌볼록집합 C 와 x_{0} \in H \setminus C 에 대하여 d = inf{\lVert x_{0} – y \rVert : y \in C} 라 두면 \lVert x_{0} – y_{0} \rVert = d 인 y_{0} \in C 가 유일하게 존재한다.

(정의) 내적공간의 두 벡터 x,y 가 \langle x, y \rangle = 0 일 때, x 와 y가 서로 수직이라고 하고 x \perp y 라 쓴다. 부분집합 E의 임의의 벡터와 수직인 벡터를 모두 모은 집합을 E^{\perp} 라 표시한다.

(따름정리 6.3.2) 힐버트공간 H의 부분공간 E가 닫힌 진부분공간이라면 E^{\perp} \supsetneqq {0} 이다.

(정리 6.3.3) 힐버트공간 H 에서 임의의 y \in H 에 대하여 \phi_{y} : X \mapsto \langle x,y \rangle , x \in H, y \mapsto \phi_{y} : X \to X^{*} 에 의해 정의된 사상 y \mapsto \phi_{y} 는 전단사 등거리 켤레선형사상이다.

(정의) 내적공간 H의 부분집합 {u_{\alpha} : \alpha \in A} 가 다음 조건을 만족할 때, 이를 정규직교집합이라 부르고, 이에 의하여 생성된 H의 부분공간을 P_{A} 라 쓰자.

\langle u_{\alpha} , u_{\beta} \rangle = \begin{cases} 1 & \alpha = \beta \ 0 & \alpha \neq \beta \end{cases}

(명제 6.3.4) 내적공간 H의 정규직교집합 {u_{\alpha} : \alpha \in A} 의 유한부분집합 {u_{\alpha} : \alpha \in F} 가 주어져 있다. 이 때, 임의의 고정된 x \in H 에 대하여 부등식 \bigg| x - \sum_{\alpha \in F} \langle x, u_{\alpha} \rangle u_{\alpha} \bigg| \le | x – y | , y \in P_{F} 이 성립하고, y = \sum_{\alpha \in F} \langle x, u_{\alpha} \rangle u_{\alpha} 일 때에 한하여 등호가 성립한다.

또한, 각 x \in H 에 대하여 부등식 \sum_{\alpha \in F} | \langle x, u_{\alpha} \rangle |^{2} \le | x |^{2} 이 성립한다.

(베셀 부등식) (따름정리 6.3.5) 내적공간 H에 주어진 임의의 정규직교집합 {u_{\alpha} : \alpha \in A } 에 대하여, 부등식 \sum_{\alpha \in A} | \langle x, u_{\alpha} \rangle |^{2} \le | x |^{2}, x \in H 이 성립한다.

(정리 6.3.6) 힐버트공간 H 의 정규직교집합 \mathfrak{U} = {u_{\alpha} : \alpha \in A } 에 대하여 다음은 동치이다.

각 x \in H 에 대하여 등식 \sum_{\alpha \in A} | \langle x, u_{\alpha} \rangle |^{2} = | x |^{2} 이 성립한다.

각 x, y \in H 에 대하여 등식 \sum_{\alpha \in A} \langle x, u_{\alpha} \rangle \bar{\langle y, u_{\alpha} \rangle} = \langle x,y \rangle 이 성립한다.

만일 \mathfrak{V} \supset \mathfrak{U} 인 정규직교집합 \mathfrak{V} 이 있으면 \mathfrak{V} = \mathfrak{U} 이다.

벡터공간 P_{A} 가 H 안에서 조밀하다.

(정의) \sum_{\alpha \in A} \langle x, u_{\alpha} \rangle \bar{\langle y, u_{\alpha} \rangle} = \langle x,y \rangle 는 파시발 등식이라 불린다.

(정의) (정리 6.3.6) 에 있는 조건들을 만족하는 정규직교집합 {u_{\alpha} : \alpha \in A } 을 힐버트공간 H 의 정규직교기저라 한다.

(정의) 힐버트공간 사이에 정의되어 내적을 보존하는 선형사상을 힐버트공간 동형사상이라 부른다.

(정리 6.3.7) 힐버트공간 H의 정규직교기저 {u_{\alpha} : \alpha \in A } 에 대하여 {Phi} :x \mapsto \hat{x} : H \to \mathcal{l}^{2} (A) 는 전단사 힐버트공간 동형사상이다.

(명제) u_{n}(t) = e^{int} 로 정의한 {u_{n} : n \in \mathbb{Z}} 는 L^{2}(\mathbb{T}) 의 정규직교기저가 된다.

등식 \sum_{n= -\infty}^{\infty} \hat{f}(n) \bar{\hat{g}(n)} = \frac{1}{2 \pi} \int_{-\pi}^{\pi} f(t) \bar{g(t)} dt, f, g \in L^{2}(\mathbb{T}) 도 역시 파시발 등식이라 부른다.

등식 \lim_{n \to \infty} \bigg| f - \sum_{k = -n}^{ n} \hat{f}(k) u_{k}\ bigg|_{2} = 0, f \in L^{2}(\mathbb{T})

실수축 \mathbb{R} 의 원래 르벡측도에 \frac{1}{2 \pi} 한 측도공간을 \hat{\mathbb{R}} 이라 쓰자. 그러면 등식 \Int_{\mathbb{R}} |f(x)|^{2} dx = \Int_{\mathbb{\hat{R}}} |\hat{f}(\alpha)|^{2} d\alpha, f \in C_{c}(\mathbb{R}) 이 성립한다. 역변환공식은 물론 f(x) = \Int_{\mathbb{\hat{R}}} \hat{f}(\alpha) e^{i \alpha x} d\alpha 로 표현된다.

(명제) 함수공간 C_{c}(\mathbb{R}) 을 L^{2} (\mathbb{R}) 의 부분공간으로 생각하면, 등거리 선형사상 f \mapsto \hat{f} : C_{c} (\mathbb{R}) \to L^{2} (\hat{\mathbb{R}}) 을 얻게 된다. 그런데 C_{c}(\mathbb{R}) 이 L^{2} (\mathbb{R}) 안에서 조밀하므로 이 사상을 L^{2}(\mathbb{R}) 에 확장하여 등거리 선형사상 \mathcal{F} : L^{2}(\mathbb{R}) \to L^{2}(\hat{\mathbb{R}}) 을 얻을 수 있다. 만일 f \L^{1}(\mathbb{R}) \cap L^{2}(\mathbb{R}) 이면 두 사상이 동시에 정의되며, \hat{f} = \mathcal{F} (f) 이다.

(정의) 위 명제에서 정의된 \mathcal{F} : L^{2}(\mathbb{R}) \to L^{2}(\hat{\mathbb{R}}) 는 전단사 등거리사상이며 이를 플랑셰를 변환이라 한다.

(정리 6.3.8) 각 f \in L^{1}(\mathbb{R}) \cap L^{2}(\mathbb{R}) 에 대하여 \hat{f} = \mathcal{F} (f) 를 만족하는 힐버트공간 동형사상 \mathcal{F} : L^{2}(\mathbb{R}) \to L^{2}(\hat{\mathbb{R}}) 이 유일하게 존재한다.

(정의) 플랑셰를 변환이 힐버트공간 동형사상이란 말을 수식으로 쓰면 \Int_{\mathbb{R}} f(x) \bar{g(x)}dx = \frac{1}{2\pi} \Int_{\mathbb{R}} \hat{f}(\alpha)\bar{\hat{g}(\alpha)} d\alpha 가 되는데, 이 역시 파시발 등식이라 불린다.

복소측도

(Section 7) 복소측도

(subsection 7.1) 부호측도와 복소측도

(정의) 만일 함수 \lambda : \mathfrak{S} \to \mathbb{R}^{*} 가 다음 성질을 만족하고 \mu(\empty) = 0 이면, 이를 부호가 붙은 측도라 한다.

{A_{n} \in \mathfrak{S} : n = 1,2, …} 가 서로소 \implies \lambda \biggl \bigsqcup_{n=1}^{\infty} A_{n} \biggr = \sum_{n = 1}^{\infty} \lambda (A_{n})

(정의) 만일 함수 \lambda : \mathfrak{S} \to \mathbb{C}가 다음 성질을 만족하고 \mu(\empty) = 0 이면, 이를 복소측도라 한다.

{A_{n} \in \mathfrak{S} : n = 1,2, …} 가 서로소 \implies \lambda \biggl \bigsqcup_{n=1}^{\infty} A_{n} \biggr = \sum_{n = 1}^{\infty} \lambda (A_{n})

(명제) 복소측도의 합과 스칼라곱은 다시 복소측도가 된다.

(정의) 복소측도 \lambda : \mathfrak{S} \to \mathbb{C} 가 주어졌을 떄, 각 E \in \mathfrak{S} 에 대하여 \lambda_{1} (E) 와 \lambda_{2} (E) 를 각각 \lambda (E) \in \mathbb{C}의 실수부분과 허수부분으로 정의하면, \lambda_{1} 와 \lambda_{2} 는 부호측도가 되고 \lambda = \lambda_{1} + i \lambda_{2} 이다.

(정의) 집합 X의 \sigma -대수 \mathfrak{S} 에 부호측도 \lambda 가 정의되어 있을 때, 집합 P \in \mathfrak{S} 가 성질 E \in \mathfrak{S} , E \subset P \implies \lambda (E) \ge 0 을 만족하면 이를 양집합이라 부르고, 성질 E \in \mathfrak{S} , E \subset P \implies \lambda (E) \le 0 을 만족하면 이를 음집합이라 정의한다.

(명제) 양집합의 부분집합은 항상 양집합이다. 만일 {P_{n} : n = 1,2, …} 이 양집합들의 모임이면 그 합집합 P 도 양집합이다.

(한 분해정리) (정리 7.1.1) 만일 \sigma -대수 (X,\mathfrak{S}) 에 부호측도 \lambda 가 주어지면 X 는 양집합과 음집합으로 분할된다. 만일 두 가지 분할 (P_{1}, Q_{1}) 과 (P_{2}, Q_{2}) 가 있으면, 임의의 E \in \mathfrak{S} 에 대하여 \lambda(E \cap P_{1}) = \lambda(E \cap P_{2}), \lambda(E \cap Q_{1}) = \lambda(E \cap Q_{2}) 가 성립한다.

(정의) 이제, 부호측도 \lambda 에 의하여 X 를 양집합 P 와 음집합 Q 로 분할하였을 때, 각 E \in \mathfrak{S} 에 대하여 \lambda_{+}(E) = \lambda(E \cap P), \lambda_{-}(E) = -\lambda(E \cap Q) 라 정의할 수 있다. 그러면 \lambda_{+}(E) 와 \lambda_{-}(E) 는 양측도가 되고 \lambda (E) = \lambda_{+}(E) - \lambda_{-}(E), E \in \mathfrak{S} 가 된다.

또한, 각 E \in \mathfrak{S} 에 대하여 | \lambda |(E) = \lambda_{+}(E) + \lambda_{-}(E)라 정의하면 | \lambda | 도 양측도이다.

(명제 7.1.2) 집합 X의 \sigma -대수 \mathfrak{S} 에서 정의된 부호측도 \lambda가 주어지면, 각 E \in \mathfrak{S} 에 대하여 | \lambda |(E) = \sup \biggl{ \sum_{n = 1}^{\infty} | \lambda (E_{n}) | : E_{n} \in \mathfrak{S} , E \bigsqcup_{n = 1}^{\infty} E_{n}\biggr} 이 성립한다.

(정의) \lambda 가 \sigma -대수 (X, \mathfrak{S}) 의 복소측도일 때 | \lambda |(E) = \sup \biggl{ \sum_{n = 1}^{\infty} | \lambda (E_{n}) | : E_{n} \in \mathfrak{S} , E \bigsqcup_{n = 1}^{\infty} E_{n}\biggr}

(정의) 집합 X의 \sigma -대수 \mathfrak{S} 에서 정의된 복소측도 \lambda 가 주어져 있을 때, E \mapsto |\lambda|(E) 는 양측도가 된다.

(정의) 복소측도 \lambda 를 \lambda = (\lambda_{1+} - \lambda_{1-}) + i ( \lambda_{2+} - \lambda_{2-} ) 와 같이 양측도의 선형결합으로 나타낼 수 있다. 임의의 E \in \mathfrak{S} 에 대하여 | \lambda | (E) < \infty 이며 특히 | \lambda | (X) < \infty 인데, \lVert \lambda \rVert = | \lambda | (X) 라 정의한다. 전체측도 \mu(X) 값이 유한인 양측도를 유한 양측도라 정의하는데, 유한 양측도는 복소측도이나 양측도는 복소측도가 아니다.

(subsection 7.2) 라돈-니코딤 정리와 르벡 분해정리

(정의) 집합 X 위의 \sigma -대수 \mathfrak{S} 위의 양측도 \mu 와 부호측도 혹은 복소측도 \lambda 가 성질 E \in \mathfrak(S), \mu(E) = 0 \implies \lambda(E) = 0 를 만족하면 \lambda 를 \mu 에 대한 절대연속측도라 하고 \lambda \ll \mu 라 쓴다.

또한 성질 \lambda (E) = \lambda (A \cap E) , \mu (E) = \mu (B \cap E) , E \in \mathfrak{S} 를 만족하는 서로소인 A, B \in \mathfrak{S} 가 있으면 \lambda 를 \mu 에 대한 특이측도라 말하며, 이 때 \lambda \perp \mu 라 쓴다.

(명제) 실수축 위에서 \lambda = \lambda_{1} + i \lambda_{2} 와 같이 정의된 측도는 르벡측도에 대하여 절대연속이다. 디락측도는 르벡측도에 대한 특이측도이다.

(명제) 양측도 \mu 에 대하여 절대연속인 두 복소측도의 선형결합은 다시 절대연속이다. \mu 에 대한 특이측도의 선형결합도 다시 특이측도이다. 만일 복소측도가 양측도 \mu 에 대하여 동시에 절대연속이며 특이측도라 하면 \lambda = 0 이다.

(정의) \sigma-대수 (X, \mathfrak{S}) 에 고정된 양측도 \mu 가 주어져 있을 때, 임의의 복소측도를 절대연속측도와 특이측도의 합으로 표시할 수 있는데, 이를 르벡 분해라 부른다.

(정의) 르벡분해의 절대연속부분은 함수로 표현되는데, 이 함수를 \mu 에 대한 \lambda 의 라돈-니코딤 도함수라 부른다.

(르벡-라돈-니코딤)(정리 7.2.1) 집합 X 의 \sigma -대수 \mathfrak{S} 위에서 정의된 \sigma -유한 양측도 \mu 와 복소측도 \lambda 에 대하여 성질 \lambda = \lambda_{a} + \lambda_{s} , \lambda_{a} \ll \mu , \lambda_{s} \perp \mu 를 가지는 측도 \lambda_{a} , \lambda_{s} 가 유일하게 존재한다. 또한 다음 등식 \lambda_{a} (E) = \int_{E} hd \mu , E \in \mathfrak{S} 이 성립하는 h \in L^{1}(X, \mu ) 가 유일하게 존재한다. 만일 \mu 와 \lambda 가 \sigma -유한 양측도이면 성질 \lambda = \lambda_{a} + \lambda_{s} 을 만족하는 측도 \lambda_{a} 와 \lambda_{s} , 그리고 등식 \lambda_{a} (E) = \int_{E} hd \mu , E \in \mathfrak{S}를 만족하는 잴 수 있는 함수 h : X \to [0,\infty] 가 유일하게 존재한다.

(정의) 두 측도 lambda_{a} 와 \mu 가 관계식 \lambda_{a} (E) = \int_{E} hd \mu , E \in \mathfrak{S} 를 만족할 때, 이를 d \lambda_{a} = h d \mu , h = \bigg[ \frac{d \lambda_{a}}{d \mu} \bigg] 로 쓴다.

(극형식분해) (정리 7.2.2) 집합 X에서 정의된 임의의 복소측도 \lambda 에 대하여 성질 d \lambda = hd| \lambda | , |h| = 1 을 만족하는 h \in L^{1}(X,| \lambda |) 가 유일하게 존재한다.

(정의) 복소함수 f \in L^{1}(X, |\lambda|) 의 복소측도 에 관한 적분은, 집합 X에 정의된 복소측도 \lambda 의 극형식 분해 d \lambda = hd| \lambda | 가 주어졌을 때, 등식 \int_{X} \lambda d \mu = (\int_{X} \lambda_{1+} d \mu - \int_{X} \lambda_{1-}) d \mu + i ( \int_{X} \lambda_{2+} d \mu - \int_{X} \lambda_{2-} d \mu ) 로 주어진다.

(정리 7.2.3) 집합 X 의 \sigma -대수에서 정의된 \sigma -유한 양측도 \mu 에 대하여 L^{p}( \mu )^{*} = L^{q}( \mu ) 이다. (단, p, q 는 켤레지수이고, 1 \le p < \infty )

(따름정리 7.2.4) 집합 X 에 \sigma -유한 양측도 \mu 가 주어져 있고, 1 \le p \le \infty 라 하자. 만일 X에서 정의된 잴 수 있는 함수 g 가 성질 h \in L^{q}( \mu ) \implies \bigg| \int_{X} ghd \mu \bigg| \le M | h |{q} (단, p, q 는 켤레지수) 을 만족하면 g \in L^{p}( \mu ) 이고 | g |{p} \le M 을 만족한다.

(subsection 7.3) 실직선의 보렐측도

(정의) 보렐집합 위에서 정의된 복소측도를 보렐측도라 한다. 유계구간의 측도값이 항상 유한인 양측도인 보렐측도를 정규 양보렐측도라 부른다.

(정의) 실직선의 정규 양보렐측도 \lambda 에 대하여, 함수 \alpha_{\lambda} : \mathbb{R} \to \mathbb{R} 을 다음과 같이 정의한다. \alpha_{\lambda} 는 단조증가이고 오른쪽연속함수이다.

\alpha_{\lambda} = \begin{cases} \lambda (( 0, x \rbrack ), & x >0 \ 0, & x = 0 \ - \lambda (( x, 0 \rbrack ), & x < 0 \end {cases}

(정의) 구간 [a,b] 위에서 정의된 유계실함수 f : [a,b] \to \mathbb{R} 에 대하여 리만-스틸체스 적분 \int_{a}^{b} fd \alpha_{\lambda} 를 생각한다. 이는 리만적분과 비슷한 과정을 거쳐 정의된다.

(명제) 함수 f : \mathbb{R} \to \mathbb{R} 이 연속이고 그 받침이 유계구간 (A,B) 에 들어간다면, 등식 \int_{\mathbb{R}} f d \lambda = \int_{A}^{B} f(x) d \alpha_{\lambda} (x) 이다. 따라서 함수공간 C_{c}(\mathbb{R}) 에서 정규 양보렐측도에관한 적분은 리만-스틸체스 적분과 같다.

(정의) 한 점의 측도값이 항상 0 인 측도를 연속측도라 부른다.

(명제 7.3.1) 실직선에서 정의된 정규 양보렐측도 \lambda 에 대하여, 함수 \alpha_{\lambda} 가 점 x 에서 연속일 필요충분조건은 \lambda({x}) = 0 이다.

(정리 7.3.2) 실직선에 정의된 정규 양보렐측도 \lambda 에 대하여 다음은 동치이다.

\lambda \ll m 이다.

\alpha_{\lambda} 가 임의의 유계닫힌구간 위에서 절대연속이다.

(정리 7.3.3) 실직선 위에서 정의된 정규 양보렐측도 \lambda 는 임의의 보렐집합 E \in \mathfrak{B} 에 대하여 다음 등식들을 만족한다.

\lambda(E) = \inf{\lambda(U) : U \supset E, U 는 열린집합이다.}

\lambda(E) = \sup{\lambda(K) : K \subset E, K 는 유계닫힌집합이다.}

(따름정리 7.3.4) 만일 \lambda \ll m 이고 거의 모든 점에서 \alpha_{\lambda} ‘ = 0 이면 \lambda = 0 이다.

(정리 7.3.5) 실직선에 정의된 정규 양보렐측도 \lambda 에 대하여 다음은 동치이다.

\lambda \perp m 이다.

거의 모든 점에서 \alpha_{\lambda} ‘ = 0 이다.

(따름정리 7.3.6) 실직선 위의 정규 양보렐측도 \lambda 가 르벡측도 m 에 대하여 절대연속이면 거의 모든 점에서 [\frac{d \lambda }{d m}] = \alpha_{\lambda}’ 이 성립한다.

(명제) 거의 모든 점에서 도함수가 0 이지만 상수함수가 아닌 연속함수가 존재하며 칸토르 함수가 대표적이다.

(명제) 실직선 위에 정의된 정규 양보렐측도 \lambda 는 절대연속측도 \lamba_{a}, 연속특이측도 \lambda_{s}, 디락측도의 합으로 표현되는 측도 \lambda_{d} 의 합으로 쓸 수 있다. 이들 측도엣 얻어지는 단조증가함수를 각각 \alpha_{a} , \alpha_{s}, \alpha_{d} 라 두면 \alpha_{a} 는 절대연속함수, \alpha_{s} 는 그 도함수가 거의 모든 점에서 0인 연속함수, \alpha_{d} 는 불연속함수이다. \lambda 에 의하여 얻어지는 단조증가함수를 \alpha 라 두면 \alpha = \alpha_{a} + \alpha_{s} + \alpha_{d} 이고 등식 \lambda_{a} (E) = \int_{E} \alpha_{a} ‘ dm = \int_{E} \alpha ‘ dm , E \in \mathfrak{B} 이 성립한다.

위상과 측도

(section 8) 위상과 측도

(subsection 8.1) 국소옹골공간

(정의) 집합 X의 부분집합 모임 \mathfrak{T} 가 다음 성질들

\empty , X \in \mathfrak{T}

\mathfrak{T} 는 임의의 합집합에 대하여 닫혀있다.

\mathfrak{T}는 유한교집합에 대하여 닫혀있다.

을 만족할 때 이 \mathfrak{T} 를 X 의 위상이라 하고, 위상이 주어진 집합을 위상공간이라 한다. \mathfrak{T} 에 들어가는 집합을 열린집합이라 하며, 그 여집합이 \mathfrak{T} 에 들어가는 집합을 닫힌집합이라 부른다. 임의의 집합 S \subset X 에 대하여 S를 포함하는 모든 닫힌 부분집합들의 교집합을 \bar{S} 라 쓰고 이를 S의 닫힘이라 한다. S 안에 들어가는 열린집합 전체의 합집합을 S의 내부라 하고 이를 int S 라 쓴다.

(정의) 모든 부분집합을 다 모으면 위상이 되는데, 이러한 위상이 주어진 위상공간을 이산공간이라 한다. 위상공간 (X, \mathfrak{T}) 와 X의 부분집합 Y \subset X 가 있을 때 \mathfrak{T} |{Y} = {A \cap Y : A \in \mathfrak{T} } 는 Y 의 위상이 되는데, (Y , \mathfrak{T} |{Y}) 를 (X, \mathfrak{T}) 의 부분공간이라 한다.

(정의) 위상공간 X, Y 사이에 정의된 함수 f : X \to Y 가 있을 떄, 임의의 열린 집합 V \subset Y 에 대하여 그 역상 f^{-1} (V) 가 X의 열린 집합이면 f 를 연속함수라 한다. 두 위상공간 X< Y 사이에 전단사함수 f 가 존재하며 f 와 f^{-1} 모두 연속 이면 X와 Y가 서로 위상동형이라 한다.

(정의) 위상공간 (X, \mathfrak{T}) 의 모든 열리집합이 집합모임 \mathfrak{B} \subset \mathfrak{T} 에 있는 집합들의 합집합으로 표시되면 \mathfrak{B} 를 이 공간의 위상기저 또는 기저라 한다.

(정의) 집합모임 {X_{\alpha} : \alpha \in A }의 곱집합을 \prod_{\alpha \in A} X_{\alpha} = {\xi : A \to \bigcup_{\alpha \in A} X_{\alpha} : \xi (\alpha ) \in X_{\alpha} (\alpha \in A)} 라 정의한다. 각 \alpha \in A 에 대하여 \pi_{\alpha} (\xi ) = \xi ( \alpha ) , \xi \in \prod_{\alpha \in A} X_{\alpha} 와 같이 정의된 함수 \pi_{\alpha} : \prod_{\alpha} X_{\alpha} \to X_{\alpha} 를 정사영이라 한다. 위상공간 모임 {(X_{\alpha}, \mathfrak{T}_{\alpha}): \alpha \in A} 에 대하여, …(p215)

(subsection 8.2) 리쓰 표현정리

(정의) 위상공간 X의 열린집합으로 포함하는 최소의 \sigma -대수 \mathfrak{B} 에 들어가는 집합을 보렐집합이라 하고, (X, \mathfrak{B}) 에 정의된 측도를 보렐측도라 한다. 보렐측도에 관하여 잴 수 있는 함수를 보렐함수라 한다.

(정의) 만일 국소옹골공간 X 에 주어진 양보렐측도 \mu 가 성질 K 가 옹골집합 \implies \mu (K) < \infty 를 만족하면 임의의 f \in C_{c}(X) 에 대하여 \int_{X} f d \mu \in \mathbb{C} 이고, \Lambda {\mu} : f \mapsto \int{X} fd \mu : C_{c} (X) \to \mathbb{C} 는 C_{c} (X) , f \ge 0 \implies \Lambda_{\mu} (f) \ge 0 이 성립하는데, 이러한 C_{c} (X) 의 선형범함수를 양선형범함수라 부른다.

만일 \Lambda 가 C_{c}(X) 의 양선형범함수이면, g, h \in C_{c}(X) 가 실함수일 때 g \le h \implies \Lambda (g) \le \Lambda (h) 이다.

(정의) X가 국소옹골공간이라 가정하고, C_{c} (X) 의 양선형범함수 \Lambda : C_{c} (X) \to \mathbb{C} 가 주어져 있다고 하자. 임의의 열린집합 V \subset X 에 대하여 \mu (V) = \sup{\lambda (f) : f \prec V} 라 정의하고, 임의의 E \subset X 에 대하여 \mu (E) = \inf {\mu(V) : V \supset E , V 는 열린집합} 이라 정의한다. 이로써 집합 X에 정의된 외측도 \mu : \mathcal{P} (X) \to [0, \infty ] 를 얻었다. 양측도 (X, \mathfrak{S}, \mu) 를 얻을 수 있다. \mathfrak{B} \subset \mathfrak{S} 임을 알 수 있고 \mu 를 \mathfrak{B} 에 제한하면 보렐측도를 얻는다.

(도움정리 8.2.1) 지금까지 만든 보렐측도 \mu 는 다음 성질을 가진다.

K가 옹골집합이면 \mu (K) < \infty 이고, 다음 등식 \mu (K) = \inf {\lambda (f) : K \prec f} 이성립한다.

임의의 열린집합 E가 다음 등식 \mu (E) = \sup {\mu(K) : K \supset E , K 는 옹골집합} 을 만족한다.

(리쓰 표현정리)(정리 8.2.2) 국소옹골 하우스도르프 공간 X가 주어져 있을 때, 벡터공간 C_{c} (X) 에서 정의된 임의의 양선형범함수 \Lambda 에 대하여 다음 성질들

임의의 f \in C_{c} (X) 에 대하여 \Lambda (f) = \int_{X} fd \mu 이다.

임의의 보렐집합 E에 대하여 \mu (E) = \inf {\mu(V) : V \supset E , V 는 열린집합} 이 성립한다.

임의의 열린집합 E에 대하여 \mu (E) = \sup {\mu(K) : K \supset E , K 는 옹골집합} 이 성립한다.

를 만족하는 보렐측도 \mu 가 유일하게 존재한다. 이 때, 측도 \mu는 조건 K 가 옹골집합 \implies \mu (K) < \infty 를 만족한다. 또한, \mu(E) < \infty 인 임의의 보렐집합 E에 대하여 \mu (E) = \sup {\mu(K) : K \supset E , K 는 옹골집합} 이 성립한다.

(정의) 국소옹골공간에 정의된 양보렐측도가 조건 K 가 옹골집합 \implies \mu (K) < \infty 를 만족하고 임의의 보렐집합에 대하여 \mu (E) = \inf {\mu(V) : V \supset E , V 는 열린집합} 와 \mu (E) = \sup {\mu(K) : K \supset E , K 는 옹골집합}이 성립하면 이를 정규보렐측도라 한다.

(명제) 리쓰 표현정리에 의하여 주어진 측도가 \sigma -유한측도라면 이는 정규측도이다. 셀 수 있는 옹골부분집합의 합집합으로 표시되는 공간을 \sigma -옹골공간이라 하는데, 이 경우 리쓰 표현정리에 의해 얻어지는 측도가 정규측도가 된다.

(명제) 실직선의 정규 양보렐측도 \mu 에 대하여 단조증가하는 오른쪽연속함수 \alpha_{\mu} 를 정의하여 대응관계 \mu \mapsto \alpha_{\mu} 를 얻었고, 이 단조함수에 의하여 정의되는 C_{c} (\mathbb{R}) 의 리만-스틸체스 적분이 앞에서 \Lambda {\mu} : f \mapsto \int{X} fd \mu : C_{c} (X) \to \mathbb{C} 로 정의된 양선형사상 \Lambda_{ \mu } 임을 알았다. 이 대응관계는 일대일 대응이다.

(명제) 만일 f \in C_{c} (\mathbb{R}^{2} ) 의 받침이 R = [a,b] \times [c,d] 에 들어가면 f 는 R 위에 연속함수이므로 이중적분 \int \int_{R} f 을 생각할 수 있는데, 이는 푸비니 정리에 의해 르벡적분 \int_{\mathbb{R}^{2}} f dm_{2} 와 같은 값이다. 따라서 양선형범함수 f \mapsto \int \int_{R} f : C_{c}( \mathbb{R}^{2} ) \to \mathbb{C} 에 의하여 얻어진 보렐측도는 르벡측도 m_{2} 를 보렐집합에 제한한것이다.

(section 8.3) 연속함수와 보렐측도

(정의) 리쓰 표현정리에 의해 얻어진 국소옹골공간의 양측도를 준정규 양보렐측도라 부른다.

X 의 준정규 양보렐측도와 C_{c} (X) 의 양선형범함수는 일대일대응 관계를 가진다.

(루진) (정리 8.3.1) 국소옹골공간 X에 준정규 양보렐측도 \mu 와 잴 수 있는 함수 f : X \to \mathbb{C} 가 주어져 있고, f 는 유한측도집합 A 바깥에서 0 이라 가정하자. 이 때, 임의의 양수 \epsilon > 0 에 대하여 \mu({x \in X : f (x) \neq g (x) }) < \epsilon 을 만족하는 연속함수 g \in C_{c} (X) 가 존재한다. 만일 f \in L^{\infty} (X, \mu) 이면 | g |{\infty} \le | f |{\infty} 가 되도록 g \in C_{c} (X) 를 택할 수 있다.

(정리 8.3.2) 국소옹골공간 X에 준정규 양보렐측도 \mu 가 주어져 있을 때, C_{c} (X) 는 L^{p} (X, \mu) 의 조밀한 부분공간이다 (단, 1 \le p < \infty)

(정의) 국소옹골공간 X에서 정의된 복소 정규보렐측도 전체의 벡터공간을 M(X) 라 쓰면 이는 \lambda \mapsto | \lambda | 에 관하여 노음공간이 된다.

(도움정리 8.3.3) 실노음공간 C_{0}^{\mathbb{R}} (X) 에서 정의된 유계 실선형범함수 \phi 에 대하여 \phi = \phi_{+} - \phi_{-} , | \phi | = | \phi_{+} | + | \phi_{-} | 인 양선형범함수 \phi_{+}, \phi_{-} 가 존재한다.

(정리 8.3.4) 국소옹골공간 X에 대하여 C_{0} (X)^{*} = M(X) 이다.

(명제 8.3.5) 국소옹골공간 X의 실함수공간 C_{c}^{\mathbb{R}} (X) 의 실부분공간 A 가 스톤-바이에르쉬티라스 정리의 첫번째와 두번째 가정을 만족한다고 가정하자. 또한, 임의의 옹골집합 K 에 대하여 K \prec h 인 h \in A 가 존재한다고 가정하자. 그러면 A는 노음공간 (C_{c}^{\mathbb{R}} (X), | |_{\sup}) 안에서 조밀하다.

(정의 8.18) X에서 정의된 보렐함수 f와 Y에서 정의된 보렐함수 g가 주어지면 X \times Y 에서 정의된 함수 (x,y) \mapsto f(x), (x,y) \mapsto g(x) 는 각각 \mathfrak{B}{X} \times \mathfrak{B}{Y} 에서 잴 수 있는 함수이고, 따라서 함수 (x,y) \mapsto f(x)g(y) , (x,y) \in X \times Y 는 \mathfrak{B}{X} \times \mathfrak{B}{Y} 에 관하여 잴 수 있는 함수이다. 만일 f \in C_{c} (X) 이고 g \in C_{c} (Y) 이면 위 처럼 정의된 함수는 C_{c} (X,Y) 에 들어가는데, 이런 형태의 함수들에 의해 생성된 C_{c} (X \times Y) 의 부분공간을 \mathcal{A} 라 두자.

(정의) 만일 \mu, \nu 가 각각 X, Y 의 \sigma -유한 준정규 양보렐측도라면, 임의의 F \in C_{c}(X \times Y) 는 L^{1} (\mu \times \nu) 에 들어간다. 따라서 C_{c} (X \times Y)의 양선형범함수 F \mapsto \int_{ X \times Y} F d( \mu \times \nu ) , F \in C_{c} (X \times Y ) 를 생각할 수 있다. 이 선형범함수에 대응하는 준정규 양보렐측도는 \sigma – 대수 \mathfrak{B}{X} \times \mathfrak{B}{Y} 위에서 곱측도 \mu \times \nu 와 그 측도값이 일치하고, X \times Y 에서 정의된 보렐함수에 대하여 토넬리-푸비니 정리가 성립한다.

(정의) (정의 8.18) 처럼 정의된 함수를 f \otimes g 라 표시하자. 만일 \mu 와 \nu 가 각각 국소옹골공간 X와 Y의 복소 정규보렐측도이면, F \mapsto \int_{X \times Y} F \cdot (h \otimes k) d(| \mu | \times | \nu |), F \in C_{0} (X \times Y) 는 C_{0} (X \times Y)의 유계선형범함수가 되고, 이에 대응하는 복소정규보렐측도를 \mu \times \nu 라 쓴다.

| \mu \times \nu | \le | \mu | | \nu |

(명제) 만일 F \in C_{0} (X \times Y) 이면 임의의 x \in X 에 대하여, F_{x} \in C_{0} (Y) 이고, 함수 x \mapsto \int_{Y} F_{x} d \nu : X \to \mathbb{C} 가 C_{0} (X) 에 들어간다.

(명제) 임의의 복소정규보렐측도 \mu \in M(X) 와 \nu \in M(Y) 에 대하여 푸비니 정리 \int_{X \times Y} F d ( \mu \times \nu ) \int_{X} \int_{Y} F(x,y) d \nu (y) d \mu (x) , F \in C_{0} (X \times Y) 가 성립한다.

삼각급수

(section 9) 삼각급수

(subsection 9.1) 푸리에-스틸체스 변환

(정의) 주기함수의 푸리에변환 f \mapsto \hat{f} : L^{1}(\mathbb{T}) \to c_{0}(\mathbb{Z}) 는 노음감소 단사 중동형이다. 그 치역은 c_{0}(\mathbb{Z}) 의 점별곱하기에 대하여 닫혀있는데, 이를 A(\mathbb{Z}) 라 쓰고 푸리에대수라 부른다. 또한 \hat{f} \in \mathcal{l}^{1}(\mathbb{Z}) 인 f \in L^{1}(\mathbb{T}) 전체의 집합을 A(\mathbb{T}) 라 쓰자. 그러면 임의의 f \in A(\mathbb{T}) 가 연속함수임을 알고 있으므로. 이를 요약하면 다음과 같다.

\begin{matrix} f & \in & A(\mathbb{T}) & \subset & C(\mathbb{T}) & \subset & \L^{2}(\mathbb{T}) & \subset & \L^{1}(\mathbb{T}) & & \ \updownarrow & & \updownarrow & & \updownarrow & &\updownarrow & & \updownarrow & & \ \hat{f} &\in & \mathcal{l}^{1}(\mathbb{Z}) & \subset & \mathfrak{X} & \subset & \mathcal{l}^{2}(\mathbb{Z}) & \subset & A(\mathbb{Z}) & \subset & c_{0} (\mathbb{Z}) \end{matrix}

대응관계 L^{2}(\mathbb{T}) \leftrightarrow \mathcal{l}^{2}(\mathbb{Z}) 는 내적이 보존되는 힐버트공간 동형사상이다.

(정의) 대응관계 A(\mathbb{T}) \leftrightarrow \mathcal{l}^{1}(\mathbb{Z}) 를 살펴보자. 각 a \in \mathcal{l}^{1}(\mathbb{Z}) 에 대하여 \hat{a} (t) = \sum_{n = -\infty}^{\infty} a(n)e^{-int} , t \in \mathbb{T} 라 정의하면 \hat{a} 는 연속함수가 되므로 선형사상 a \mapsto \hat{a} : \mathcal{l}^{1} (\mathbb{Z}) \to C(\mathbb{T}) 를 얻는다. 이 사상의 치역은 A(\mathbb{T}) 가 된다. 이 선형사상은 \mathcal{l}^{1}(\mathbb{Z}) 의 푸리에변환이라 불린다.

(정의) 국소옹골공간 X에서 정의된 유계연속함수 전체의 벡터공간을 C^{b}(X) 라 쓰면 f \mapsto | f |_{sup} 는 이 공간의 노음이 된다. 만일 X가 이산공간이면 c^{b}(X) 라 쓴다. 따라서 \mu \mapsto \hat{\mu} : M(\mathbb{T}) \to c^{b}(\mathbb{Z}) 는 노음감소 선형사상이 되는데, 이를 푸리에-스틸체스 변환이라 부른다.

이 변환을 L^{1}(\mathbb{T}) 로 제한하면 푸리에 변환을 얻는다.

(정의) 두 복소측도 \mu, \nu \in M(\mathbb{T}) 에 대하여 C(\mathbb{T}) 의 선형범함수를 h \mapsto \int_{\mathbb{T} \times \mathbb{T}} h(s + t) d( \mu \times \nu)(s,t) , h \in C(\mathbb{T}) 와 같이 정의하자. 이는 C(\mathbb{T}) 의 유계선형범함수가 되는데, 이에 대응하는 복소측도를 \mu * \nu \in M(\mathbb{T}) 라 쓴다.

다음 부등식 | \mu * \nu | \le | \mu | | \nu | , \mu, \nu \in M(\mathbb{T}) 이 성립한다.

(정의) 만일 \mu, \nu \in M(\mathbb{T}) 가 절대연속측도이고 각각 f , g \in L^{1}(\mathbb{T}) 에 대하여 d \mu (s) = f(s) ds 와 d \nu (t) = g(t)dt 로 각각 표현되면, d(\mu \times \nu) = (f * g) dm 이 된다. 따라서, 측도의 곱하기 \mu * \nu 를 L^{1}(\mathbb{T}) 에 제한하면 L^{1}- 함수의 곱하기가 된다.

(명제) 정규보렐측도공간 M(\mathbb{T}) 는 항등원을 가지는 바나하대수이고, L^{1}(\mathbb{T}) 는 M(\mathbb{T}) 의 이데알이다.

L^{1}(\mathbb{T}) 는 M(\mathbb{T}) 의 이데알이라 함은, f \in L^{1}(\mathbb{T}) 이고 \mu \in M(\mathbb{T}) 이면 f * \mu \in L^{1} (\mathbb{T}) 임을 뜻한다.

(정의) \mu, \nu \in M(\mathbb{T}) 와 각 n \in \mathbb{Z}) 에 대하여 푸리에-스틸체스 변환은 M(\mathbb{T}) 에서 c^{b}(\mathbb{Z}) 로 가는 준동형이 된다. 따라서, 이 변환의 치역이 점별곱하기에 닫혀있는데, 이를 푸리에-스틸체스 대수라 부르고, B(\mathbb{Z}) 라 쓴다.

(정리 9.1.2) 푸리에-스틸체스 변환 \mu \mapsto \hat{\mu} 는 M(\mathbb{T}) 에서 c^{b}(\mathbb{Z}) 로 가는 노음감소 단사 준동형이다.

(정의) 지금부터 f \in C(\mathbb{T}) 와 \mu \in M(\mathbb{T}) 에 대하여 \langle f, \mu \rangle = \int_{\mathbb{T}} f d \mu 라 쓴다.

(명제) 임의의 f \in C(\mathbb{T}) 와 \mu \in M(\mathbb{T}) 에 대하여 \lim_{n \to \infty} \sum_{k = -n}^{n} ( 1- \frac{|k|}{n+1}) \hat{f}(k) \hat{\mu}(-k) = \lim_{n \to \infty} \langle K_{n} * f , \mu \rangle = \langle f, \mu \rangle 이다.

\langle P , \mu \rangle = \sum_{k} \hat{P} (k) \hat{\mu} (-k) , P \in T(\mathbb{T}) , \mu \in M(\mathbb{T}) 이다.

(정의) 삼각다항식은 연속함수로 이해할 수도 있지만 측도로 이해할 수도 있다. 삼각다항식 P를 측도로 이해하는 경우 그 노음을 | P |_{M(\mathbb{T})} 라 쓴다.

(정리 9.1.3) 함수 a : \mathbb{Z} \to \mathbb{C} 에 대하여 다음은 동치이다.

a \in B(\mathbb{Z}) 이다.

임의의 P \in T(\mathbb{T}) 에 대하여 |\sum_{k} \hat{P}(k) a(-k)| \le M | P |_{sup} 이 성립하는 상수 M 이 존재한다.

M’ := \sup_{n \in \mathbb{N}} | \sum_{k = -n}^{n} (1-\frac{|k|}{n+1}) a(k) u_{k} |_{M(\mathbb{T}} < \infty 이다.

(명제) a \in B(\mathbb{Z}) 이고 \hat{\mu} = a 인 경우, d \mu_{n} = [\sum_{k = -n}^{n} (1-\frac{|k|}{n+1}) \hat{\mu}(k) u_{k}] dm 으로 정의된 측도 \mu_{n} 을 생각하면 \lim_{n \to \infty} \langle f, \mu_{n} \rangle = \langle f, \mu \rangle, f \in C(\mathbb{T}) 를 얻는다. 측도는 C(\mathbb{T}) 에서 정의된 선형범함수이므로, 이는 측도열 \langle \mu_{n} \rangle 이 \mu 로 점별수렴함을 말한다.

(정의) 함수 a : \mathbb{Z} \to \mathbb{C} 가 성질 c_{1}, c_{2}, …, c_{k} \in \mathbb{C}, n_{1}, …, n_{k} \in \mathbb{Z} \implies \sum_{i,j =1}^{k} a(n_{i} – n_{j}) c_{i}\hat{c}_{j} \ge 0 을 만족하면 이를 양부호수열이하 부른다.

(헬그로츠)(정리 9.1.4) 함수 a : \mathbb{Z} \to \mathbb{C} 가 양부호수열일 필요층분조건은 a가 양측도의 푸리에-스틸체스 변환임이다.

(정의) 정수집합에서 정의된 양부호수열 전체의 집합을 P(\mathbb{Z}) 라 쓴다. 이 때, P(\mathbb{Z}) 가 더하기와 양수곱에 의하여 닫혀있음은 자명하다.

임의의 \mu \in M(\mathbb{T}) 가 양측도들의 선형결합으로 쓸 수 있음을 알고 있으므로, 다음 관계 B(\mathbb{Z}) = {(a_{1} – a_{2}) + i(a_{3} – a_{4}) : a_{i} \in P(\mathbb{Z}) } 가 성립함을 알 수 있다.

(subsection 9.2) 푸리에-쉬와르츠 변환

(정의) 무한번 미분가능하고 주기가 2 \pi 인 주기함수 전체의 집합을 \mathcal{D}(\mathbb{T}) 라 둔다.

(정의) 함수 a : \mathbb{Z} \to \mathbb{C} 가 성질 \sup{|n|^{p} |a(n)| : n \in \mathbb{Z}} < \infty , p = 1,2, … 를 만족하면, 이를 빠른감소수열이라 부른다. 즉, f \in \mathcal{D} (\mathbb{T}) 이면 \hat{f} 는 빠른감소수열이다.

(정의) \mathcal{D}(\mathbb{T}) 에 거리를 정의한다. d(f,g) = \sum_{p = 0}^{\infty} \frac{2^{-p} | D^{p} f – D^{p} g |_{sup}}{1 + | D^{p} f – D^{p} g | } = , f,g \in \mathcal{D}(\mathbb{T}) 라 정의한다.

(명제 9.2.1) 벡터공간 \mathcal{D} (\mathbb{T}) 의 함수열 \langle f_{n} \rangle 과 f \in \mathcal{D} (\mathbb{T}) 에 대하여 다음은 동치이다.

\lim_{n \to \infty} d(f_{n}, f) = 0 이다.

각 p = 0,1,… 에 대하여 \langle D^{p} f_{n} \rangle 이 D^{p} f 로 고르게 수렴한다.

임의의 양수 \epsilon >0 에 대하여 n \ge N, p < \frac{1}{\epsilon} \implies | D^{p}f_{n} – D^{p} f |_{sup} < \epsilon 이 성립하는 자연수 N 이 존재한다.

(명제 9.2.2) 푸리에변환 f \mapsto \hat{f} 는 \mathcal{D}(\mathbb{T}) 에서 빨리감소하는 수열 전체의 벡터공간으로 가는 전단사대응이다. 또한 각 f \in \mathcal{D} (\mathbb{T}) 에 대하여 \lim_{n \to \infty} d(f, \sum_{k = -n}^{n} \hat{f} (k) u_{k}) = 0 이 성립한다.

(명제) d(f + h, g+h) = d(f,g) , f,g,h \in \mathcal{D} (\mathbb{T}) . 두 점을 나란히 평행이동해도 그 거리가 변하지 않는다.

\mathcal{D}(\mathbb{T}) 의 더하기가 (f_{0}, g_{0}) \in \mathcal{D}(\mathbb{T}) \times \mathcal{D}(\mathbb{T}) 에서 연속임을 알 수 있다.

(정의) 각 P = 0,1,2, … 에 대하여 | f |{P} = max{| D^{p} f |{sup} : p = 0,1,2, …, P }, f \in \mathcal{D} (\mathbb{T}) 라 정의하면 이는 벡터공간 \mathcal{D} (\mathbb{T}) 의 노음이 되므로, N_{p} = {f \in \mathcal{D} (\mathbb{T}) : | f |_{P} \le 2^{-P}} 는 볼록집합이 된다.

(정의) 위상공간의 한 점 x의 근방들의 모임 \mathcal{N} 이 있을 때, 임의의 x를 포함하는 열린집합 V 에 대하여 x \in N \subset V 가 되는 N \in \mathcal{N} 이 존재하면 \mathcal{N}을 점 x의 국소위상기저 혹은 국소기저라 부른다.

\mathcal{D} (\mathbb{T}) 의 볼록집합들의 모임 {N_{p} : P = 0,1,2,…} 는 원점의 국소기저가 된다.

(정의) 평행이동은 위상동형사상이므로, 원점의 국소기저를 평행이동하면 각 점의 국소기저를 얻을 수 있다. 원점의 국소기저를 그냥 국소기저라 부른다. 근방이란 0의 근방을 말한다.

(정의) 벡터공간 X에 평행이동에 불변인 거리가 있어서 완비공간이 되고 볼록집합으로 이루어진 국소기저를 가지면 이를 프레쉐 공간이라 부른다.

\mathcal{D} (\mathbb{T}) 는 프레쉐 공간이다. 임의의 바나하공간은 프레쉐 공간이다.

(정의) \mathcal{D} (\mathbb{T}) 에 정의된 연속선형범함수 전체의 벡터공간을 \mathcal{D}’ (\mathbb{T}) 라 쓰자. 이는 관계 P \le Q \implies (\mathcal{D} (\mathbb{T}), | |{P})^{*} \subset (\mathcal{D} (\mathbb{T}), | |{Q})^{*} \subset \mathcal{D}’ (\mathbb{T}) 이다.

(정리 9.2.3) 벡터공간 \mathcal{D} (\mathbb{T}) 의 선형범함수 \phi 에 대하여 다음은 동치이다.

\phi 가 거리공간 (\mathcal{D} (\mathbb{T}), d) 에서 \mathbb{C} 로 가는 연속함수이다.

적절한 양수 C>0 와 자연수 P 에 대하여 다음 f \in \mathcal{D} (\mathbb{T}) \implies |\phi (f)| \le C | f |_{P} 이 성립한다.

(정의) \phi \in \mathcal{D}’ (\mathbb{T}) 와 f \in \mathcal{D} (\mathbb{T}) 에 대하여 \langle f, \phi \rangle = \phi (f) 라 쓰기로 약속한다.

각 \phi \in \mathcal{D}’ (\mathbb{T}) 와 f \in \mathcal{D} (\mathbb{T}) 에 대하여 (\phi \boxtimes f) (t) = \langle f_{-t} , \phi \rangle , t \in \mathbb{T} 라 정의한다.

(정의) \phi \boxtimes f \in \mathcal{D} (\mathbb{T}) 임을 알 수 있다. 따라서, 각 \phi, \psi \in \mathcal{D} ‘ (\mathbb{T}) 에 대하여 \langle f, \phi * \psi \rangle = \langle \psi \boxtimes f , \phi \rangle , f \in \mathcal{D} (\mathbb{T}) 와 같이 정의할 수 있다.

(정의) \phi, \phi_{n} 이 \mathcal{D} (\mathbb{T}) 의 선형범함수일 때, \lim_{n \to \infty} \langle f, \phi_{n} \rangle = \langle f , \phi \rangle , f \in \mathcal{D} (\mathbb{T}) 가 성립하면 \langle \phi_{n} \rangle 이 \phi 로 수렴한다고 정의하고, 이를 \phi_{n} \to \phi 라 쓴다.

(명제 9.2.4) 만일 각 n = 1,2, … 에 대하여 \phi {n} \in \mathcal{D} ‘ (\mathbb{T}) 이고 \phi{n} \to \phi 이면 \phi \in \mathcal{D} ‘ (\mathbb{T}) 이다.

(고른유계원칙) (정리 9.2.5) 프레쉐공간 X 에서 정의된 연속선형범함수열 \langle \phi_{k} \rangle 가 주어져 있고, 각 x \in X 에 대하여 \phi (x) = \lim_{k \to \infty} \phi_{k} (x) 이면 \phi 도 연속선형범함수이다.

(정의) 각 \phi \in \mathcal{D} ‘ (\mathbb{T}) 에 대하여 \hat{\phi} (n) = \langle u_{-n}, \phi \rangle , n \in \mathbb{Z} 라 정의하자. 각 n \in \mathbb{Z} 에 대하여 \hat{\phi * \psi }(n) = \hat{\phi} (n) \hat{\psi} (n) 이 성립한다.

(정의) 함수 a : \mathbb{Z} \to \mathbb{C} 가 있을 떄, 적절한 자연수 p 에 대하여 n \mapsto |n|^{-p} a(n) 이 유계이면 이를 슬슬증가수열이라 한다.

슬슬증가수열 전체의 벡터공간이 점별곱하기에 대하여 닫혀있다.

따라서 \phi \mapsto \hat{\phi} 는 \mathcal{D} ‘ (\mathbb{T}) 에서 슬슬증가수열 전체의 집합으로 가는 준동형인데, 이를 푸리에-쉬와르츠 변환이라 한다.

(명제) 임의의 수열 a : \mathbb{Z} \to \mathbb{C} 에 대하여 삼각다항식 \phi_{n} = \sum_{k = -n}^{n} a(k) u_{k} , n = 1,2, … 라 하자. 각 m \in \mathbb{Z} 와 n = 1,2, … 에 대하여 \langle u_{m}, \phi_{n} \rangle = \sum_{k = -n}^{n} a(k) \langle u_{m}, u_{k} \rangle = \begin{cases} a(-m) & |m| \le n \ 0 & |m| > n \end{cases} 임을 알 수 있다.

(정리 9.2.6) 푸리에-쉬와르츠 변환은 \mathcal{D} ‘ (\mathbb{T}) 에서 슬슬증가하는 수열 전체의 대수로 가는 전단사 준동형이다.

(명제) ‘빨감’ , ‘슬증’ 이 빠른감소수열 과 슬슬증가수열 전체의 공간을 나타낼 때, \begin{matrix} \mathcal{D} (\mathbb{T}) & \subset & C(\mathbb{T}) & \subset & \L^{2}(\mathbb{T}) & \subset & M(\mathbb{T}) & \subset & \mathcal{D} ‘ (\mathbb{T}) \ \updownarrow & & \updownarrow & & \updownarrow & &\updownarrow & & \updownarrow \ 빨감 & \subset & \mathfrak{X} & \subset & \mathcal{l}^{2}(\mathbb{Z}) & \subset & B(\mathbb{Z}) & \subset & 슬증 \end{matrix}

로 나타낼 수 있다. C(\mathbb{T}) 의 쌍대공간이 M(\mathbb{T}) 이고, \mathcal{D} (\mathbb{T}) 의 쌍대공간이 \mathcal{D} ‘ (\mathbb{T}) 이다.

(subsection 9.3) 약위상

(정의) 프레쉐공간 X 에서 정의된 모든 연속선형범함수 전체의 벡터공간을 X^{} 라 쓰고, 기호 \langle x, \phi \rangle = \phi(x) , x \in X , \phi \in X^{} 를 그대로 유지한다.

(정의) 복소평면 \mathcal{C} 는 이미 위상공간이므로 \mathcal{C} ^{X} 도 곱위상을 생각함으로써 위상공간이 되고, X^{} 를 \mathcal{C}^{X} 의 부분위상공간으로 이해할 수 있다 따라서 X^{} 의 위상은 {\phi \in X^{} : \langle x, \phi \rangle \in U} , x \in X, 열린집합 U \subset \mathbb{C} 을 포함하는 최소의 위상인데, 이렇게 정의된 X^{} 의 위상을 약^{*}-위상이라 한다.

복소평면이 하우스도르프공간이므로 그 곱공간 \mathcal{C}^{X} 의 부분공간 X^{*} 도 하우스도르프 공간이다.

(명제) X^{} 의 약^{} -위상 은 모든 선형범함수 \Lambda_{x} (단, x \in X) 가 연속이 되도록 하는 최소의 위상이다.

(정의) 임의의 집합 A \subset X 에 대하여 A^{ \bullet } = {\phi \in X^{*} : |\phi (x)| \le 1 (x \in A) } 라 정의하자.

(명제) 각 유한집합 F \subset X 에 대하여 \phi {0} + F^{ \bullet } 을 다 모으면 이는 약^{*} -위상에 관한 \phi{0} \in X^{*} 의 국소기저가 된다. 집합모임 {F^{\bullet} : F \subset X, F 는 유한집합} 은 원점의 국소기저가 된다.

(정의) 벡터공간 의 부분집합 A, B 와 스칼라 \alpha \in \mathbb{C} 에 대하여 A + B = {x + y : x \in A, y \in B} , \alpha A = { \alpha x : x \in A } 라 정의한다. 벡터공간 X 에 위상이 주어지고 더하기와 스칼라곱이 연속이면 이를 위상벡터공간이라 한다.

힐버트공간이나 바나하공간, 프레쉐공간은 모두 위상벡터공간이 된다.

(정의) 위상벡터공간 X에서 정의된 연속선형범함수들의 벡터공간을 X^{} 라 쓰고, 이를 X의 쌍대공간이라 한다. 벡터공간 X^{} 에 약^{*} -위상이 부여된 위상벡터공간을 X^{*w} 로 표시하자.

(명제 9.3.1) 위상벡터공간 X의 쌍대공간 X^{*w} 의 그물 \langle \phi_{i} : i \in I \rangle 와 \phi 가 있을 때, \lim_{i} \phi_{i} = \phi 일 필요충분조건은 \lim_{i} \langle x, \phi_{i} \rangle = \langle x , \phi \rangle , x \in X 이다.

(정의) 위상벡터공간의 쌍대공간에 주어진 약^{*} -위상은 국소기저 F^{ \bullet } 가 모두 볼록집합이다. 이와 같이 볼록집합으로 이루어진 국소기저를 가지는 위상벡터공간을 국소볼록공간이라 한다.

(바나하-알라오글루) (정리 9.3.2) 위상벡터공간의 근방 V 에 대하여 V^{\bullet} 는 약^{*} -위상에 관하여 옹골집합이다.

(정의) 벡터공간의 볼록집합 C 와 점 x \in C 에 대하여, 다음 조건이 성립할 때 x를 C의 꼭지점이라 부른다.

y,z \in C , x = ty + (1-t)z, 0 < t < 1 \implies x = y = z

(정의) 벡터공간의 점 x가 {x_{1}, …, x_{n}} 의 볼록결합이라는 말은 x = t_{1}x_{1}, …, t_{n}x_{n} , t_{1} + … + t_{n} = 1 , t_{i} \ge 0 ( i = 1, … , n) 을 만족하는 실수 t_{1}, …, t_{n} \in [0,1] 이 존재함을 뜻한다.

(명제) (크라인-밀만정리) 국소볼록공간에서 임의의 볼록옹골집합은 항상 꼭지점을 가지고 있으며 그 꼭지점들의 볼록결합을 모두 모으면 원래 볼록집합의 조밀한 부분집합이 된다.

(명제 9.3.3) 바나하공간 X가 다른 바나하공간 Y의 쌍대공간이라 하자. 그러면 단위공 \bar{X_{1}} = {x \in X : | x | \le 1} 이 꼭지점을 가진다.

따라서 바나하공간 L^{1}(\mathbb{T}) 의 단위공에는 꼭지점이 없으므로 이로부터 L^{1}(\mathbb{T}) 에는 약^{*} -위상을 정의하는 것이 불가능하는 것을 알 수 있다.

Multivariate Analysis

Functions on Euclidean Space

(Section 1) Functions on Euclidean Space

(subsection) Norm and Inner Product

(Def) Euclidean n-space \mathbf{R}^{n} is defined as the set of all n-tuples (x^{1}, \cdots, x^{n}) of real numbers x^{i} . An element of \mathbf{R}^{n} is often called a point in \mathbf{R}^{n} , and \mathbf{R}^{1}, \mathbf{R}^{2}, \mathbf{R}^{3} are often called the line, the plane, and space, respectively. If x denotes an element of \mathbf{R}^{n}, then x is an n-tuple of numbers, the ith one of which is denoted x^{i} ;. thus we can write x = (x^{1}, \cdots, x^{n}).

(Def) a point in \mathbf{R}^{n} is also called a vector in \mathbf{R}^{n}, because \mathbf{R}^{n}, with x + y = (x^{1} + y^{1}, \cdots, x^{n} + y^{n}) and ax = (ax^{1}, \cdots, ax^{n}) , as operations, is a vector space.

(Def) The length of a vector x, the norm |x| of x and defined by \x| = \sqrt{(x^{1})^{2} + \cdots + (x^{n})^{2}} .

(Thm 1.1) If x , y \in \mathbf{R}^{n} and a \in \mathbf{R} , then

|x| \ge 0, and |x| = 0 iff x = 0

| \sum_{i = 1}^{n} x^{i} y^{i} | \le |x| \cdot |y| ; equality holds iff x and y are linearly depencent.

|x + y| \le |x| + |y| .

|ax| = |a| \cdot |x|.

(Def) The quantity \sum_{i = 1}^{n} x^{i} y^{i} is called the inner product of x and y and denoted \langle x,y \rangle.

(Thm 1.2) If x, x_{1}, x_{2} and y, y_{1}, y_{2} are vectors in \mathbf{R}^{n} and a \in \mathbf{R}, then

\langle x , y \rangle = \langle y, x \rangle (symmetry)

\langle ax , y \rangle = \langle x , ay \rangle = a \langle x , y \rangle \langle x_{1} + x_{2} , y \rangle = \langle x_{1} , y \rangle + \langle x_{2} , y \rangle , \langle x , y_{1} + y_{2} \rangle = \langle x , y_{1} \rangle + \langle x , y_{2} \rangle (bilinearity)

\langle x , x \rangle \ge 0 and \langle x , x \rangle = 0 iff x = 0 ( positive definiteness)

|x| - \sqrt{ \langle x , x \rangle }

\langle x , y \rangle = \frac{|x + y|^{2} - |x – y|^{2}}{4} (polarization identity)

(Def) The vector (0, \cdots, 0) will be denoted simply 0. The usual basis of \mathbf{R}^{n} is e_{1} , \cdots, e_{n} where e_{i} = (0, \cdots, 1, \cdots, 0) with the 1 in the ith place.

(Def) If T : \mathbf{R}^{n} \to \mathbf{R}^{m} is a linear transformation, the matrix of T with respect to the usual bases of \mathbf{R}^{n} and \mathbf{R}^{m} is the m \times n matrix A = (a_{ij}) , where T(e_{i}) = \sum_{j = 1}^{m} a_{ji}e_{j} – the coefficients of T(e_{i}) appear in the ith column of the matrix.

If S : \mathbf{R}^{m} \to \mathbf{R}^{p} has the p \times m matrix B, then S \bullet T has the p \times n matrix BA.

(Def) if x \in \mathbf{R}^{n} and y \in \mathbf{R}^{m}, then (x,y) denotes (x^{1} , \cdots, x^{n} , y^{1}, \cdots, y^{m}) \in \mathbf{R}^{n+m}

(subsection) Subsets of Euclidean Space

(Def) If A \subset \mathbf{R}^{m} and B \subset \mathbf{R}^{n}, then A \times B \subset \mathbf{R}^{m+n} is defined as the set of all (x,y) \in \mathbb{R}^{m+n} with x \in A and y \in B.

If A \subset \mathbf{R}^{m}, B \subset \mathbf{R}^{n} and C \subset \mathbf{R}^{p}, then (A \times B) \times C = A \times (B \times C) , and denoted simply A \times B \times C.

(Def) The set [a_{1}, b_{1}] \times \cdots \times [a_{n} , b_{n}] \subset \mathbf{R}^{n} is called a closed rectangle in \mathbf{R}^{n}, while the set (a_{1}, b_{1}) \times \cdots \times (a_{n} , b_{n}) \subset \mathbf{R}^{n} is called an open rectangle.

a set U \subset \mathbf{R}^{n} is called open if for each x \in U there is an open rectangle A s.t. x \in A \subset U. a subset C of \mathbf{R}^{n} is closed if \mathbf{R}^{n} – C is open.

(prop) If A \subset \mathbf{R}^{n} and x \in \mathbf{R}^{n}, then one of three possibilities must hold.

There is an open rectangle B s.t. x \in B \subset A. (Interior of A)

There is an open rectangle B s.t. x \in B \subset \mathbf{R}^{n} – A. (Exterior of A)

If B is any open rectangle with x \in B, then B contains points of both A and \mathbf{R}^{n} – A. (Boundary of A)

(Def) A collection \mathcal{O} of open sets is an open cover of A, or briefly , covers A , if every points x \inA is in some open set in the collection \mathcal{O}.

(Def) A set A is called compact if every open cover \mathcal{O} contains a finite subcollection of open sets which also covers A.

(Heine-Borel)(Thm 1.3) The closed interval [a,b] is compact.

(Thm 1.4) If B is compact and \mathcal{O} is an open cover of {x} \times B, then there is an open set U \subset \mathbf{R}^{n} containing x s.t. U \times B is covered by a finite number of sets in \mathcal{O}.

(Cor 1.5) If A \subset \mathbf{R}^{n} and B \subset \mathbf{R}^{m} are compact, then A \times B \subset \mathbf{R}^{n+m} is compact.

(Cor 1.6) A_{1} \times \cdots \times A_{k} is compact if each A_{i} is. In particular, a closed rectangle in \mathbf{R}^{k} is compact.

(Cor 1.7) A closed bounded subset of \mathbf{R}^{n} is compact. The converse is also true.

(subsection) Functions and Continuity

(Def) A function from \mathbf{R}^{n} to \mathbf{R}^{m} (a vector-valued function of n variables) is a rule which associates to each point in \mathbf{R}^{n} some point in \mathbf{R}^{m}; the point a function f associates to x is denoted f(x).

Notation f : A \to \mathbf{R}^{m} indicates that f(x) is defined only for x in the set A, which is called the domain of f.

If B \subset A, we define f(B) as the set of all f(x) for x \in B, and if C \subset \mathbf{R}^{m} we define f^{-1}(C) = {x \in A : f(x) \in C} .

(Def) If f : A \to \mathbf{R}^{m} and g : B \to \mathbf{R}^{p}, where B \subset \mathbf{R}^{m}, then the composition g \bullet f is defined by g \bullet f (x) = g(f(x)) ; the domain of g \bullet f is A \cap f^{-1}(B). If f : A \to \mathbf{R}^{m} is 1-1, we define f^{-1} : f(A) \to \mathbf{R}^{n} by the requirement that the f^{-1} (z) is the unique x \in A with f(x) = z.

(Def) A function f : A \to \mathbf{R}^{m} determines m component functions f^{1}, \cdots, f^{m} : A \to \mathbf{R} by f(x) = (f^{1}(x), \cdots, f^{m}(x)) .

(Def) If \pi : \mathbf{R}^{n} \to \mathbf{R}^{n} is the identity function, \pi(x) = x, then \pi^{i} (x) = x^{i} ; the function \pi^{i} is called the ith projection function.

(Def \lim_{x \to a} f(x) = b means, for every number \epsilon >0 there is a number \delta>0 s.t. |f(x) – b| < \epsilon for all x in the domain of f which satisfy 0< |x-a| < \delta.

(Def) A function f : A \to \mathbf{R}^{m} is called continuous at a \in A if \lim_{x \to a} f(x) = f(a), and f is simply called continuous if it is continuous at each a \in A.

(Thm 1.8) If A \subset \mathbf{R}^{n}, a function f : A \to \mathbf{R}^{m} is continuous iff for every open set U \subset \mathbf{R}^{m} there is some open set V \subset \mathbf{R}^{n} s.t. f^{-1}(U) = V \cap A.

(Thm 1.9) If f : A \to \mathbf{R}^{m} is continuous, where A \subset \mathbf{R}^{n}, and A is compact, then f(A) \subset \mathbf{R}^{m} is compact.

(Def) If f : A \to \mathbf{R} is bounded, the extent to which f fails to be continuous at a \n A can be measured in a precise way. For \delta >0 let

M(a, f, \delta) = \sup{f(x) : x \in A and |x-a| < \delta},

m(a, f, \delta) = \inf{f(x) : x \in A and |x-a| < \delta}.

The oscillation o(f,a) of f at a is defined by o(f,a) = \lim_{\delta \to 0} [M(a,f,\delta) – m(a,f,\delta)].

(Thm 1.10) The bounded function f is continuous at a iff o(f,a) = 0.

(Thm 11.1) Let A \subset \mathbf{R}^{n} be closed. If f : A \to \mathbf{R} is any bounded function, and \epsilon >0, then {x \in A : o(f,x) \ge \epsilon } is closed.

Differentiation

(Section 2) Differentiation

(subsection) Basic Definition

(Def) A function f : \mathbf{R}^{n} \to \mathbf{R}^{m} is differentiable at a \in \mathbf{R}^{n} if there is a linear transformation \lambda : \mathbf{R}^{n} \to \mathbf{R}^{m} s.t. \lim_{h \to 0} \frac{|f(a+h) – f(a) - \lambda(h)| }{|h|} = 0.

The linear transformation \lambda is denoted Df(a) and called the derivative of f at a.

(Thm 2.1) If f : \mathbf{R}^{n} \to \mathbf{R}^{m} is differentiable at a \in \mathbf{R}^{n}, there is a unique linear transformation \lambda : \mathbf{R}^{n} \to \mathbf{r}^{m} s.t. \lim_{h \to 0} \frac{|f(a+h) – f(a) - \lambda(h)| }{|h|} = 0.

(Def) The matrix of Df(a) : \mathbf{R}^{n} \to \mathbf{R}^{m} with respect to the unsual bases of \mathbf{R}^{n} and \mathbf{R}^{m}, this m \times n matrix is called the Jacobian matrix of f at a, and denoted f’(a).

(prop) a function f : \mathbf{R}^{n} \to \mathbf{R}^{m} to be differentiable on A if f is differentiable at a for each a \in A. If f : A \to \mathbf{R}^{m} , then f is called differentiable if f can be extended to a differentiable function on some open set containing A.

(subsection) Basic Theorems

(Chain Rule) (Thm 2.2) If f : \mathbf{R}^{n} \to \mathbf{R}^{m} is differentiable at a, and g : \mathbf{R}^{m} \to \mathbf{R}^{p} is differentiable at f(a), then the composition g \bullet f : \mathbf{R}^{n} \to \mathbf{R}^{p} is differentiable at a, and D(g \bullet f ) (a) = Dg(f(a)) \bullet Df(a).

(Thm 2.3)

If f : \mathbf{R}^{n} \to \mathbf{R}^{m} is a constant function (that is, if for some y \in \mathbf{R}^{m} we have f(x) = y for all x \in \mathbf{R}^{n}), then Df(a) = 0.

If f : \mathbf{R}^{n} \to \mathbf{R}^{m} is a linear transformation, then Df(a) = f.

If f : \mathbf{R}^{n} \to \mathbf{R}^{m}, then f is differentiable at a \in \mathbf{R}^{n{ iff each f^{i} is, and Df(a) = (Df^{1}(a), \cdots, Df^{m}(a)). Thus f’(a) is the m \times n matrix whose ith row is (f^{i})’(a).

If s : \mathbf{R}^{2} \to \mathbf{R} is defined by s(x,y) = x + y, then Ds(a,b) = s.

If p : \mathbf{R}^{2} \to \mathbf{R} is defined by p(x,y) = x \cdots y, then Dp(a,b)(x,y) = bx + ay. Thus p’(a,b) = (b,a).

(Cor 2.4) If f,g : \mathbf{R}^{n} \to \mathbf{R} are differentiable at a, then D(f+g)(a) = Df(a) + Dg(a), D(f \cdot g) (a) = g(a)Df(a) + f(a) Dg(a). If, moreover, g(a) \neq 0, then D(f/g)(a) = \frac{g(a)Df(a) – f(a)Dg(a)}{[g(a)]^{2}}.

(subsection) Partial Derivatives

(Def) If f : \mathbf{R}^{n} \to \mathbf{R} and a \in \mathbf{R}^{n} , the limit \lim_{h \to 0} \frac{f(a^{1}, \cdots, a^{i} +h, \cdots, a^{n}) – f(a^{1}, \cdots, a^{n}) } {h} if it exists, is denoted D_{i}f(a), and called the ith partial derivative of f at a.

(Def) If D_{i}f(x) exists for all x \in \mathbf{R}^{n}, we obtain a function D_{i}f : \mathbf{R}^{n} \to \mathbf{R} . The jth partial derivative of this function at x, that is, D_{j}(D_{i}f) (x) , is often denoted D_{i,j} f(x).

(Thm 2.5) If D_{i,j} f and D_{j,i} f are continuous in an open set containing a, then D_{i,j} f(a) = D_{j,i} f(a). The function D_{i,j} f is called a second-order (mixed) partial derivative of f.

(prop) the order of i_{1}, \cdots, i_{k} is immaterial in D_{i1, \cdots, ik} f if f has continuous partial derivatives of all orders. A function with this property is called a C^{\infty} function.

(Thm 2.6) Let A \subset \mathbf{R}^{n}. If the maximum (or minimum) of f : A \to \mathbf{R} occurs at a point a in the interior of A and D_{i}f(a) exists, then D_{i}f(a) = 0.

(subsection) Derivatives

(Thm 2.7) If f : \mathbf{R}^{n} \to \mathbf{R}^{m} is differentiable at a, then D_{j}f^{i}(a) exists for 1 \le i \le m, 1 \le j \le n and f’(a) is the m \times n matrix (D_{j}f^{i}(a)).

(Thm 2.8) If f : \mathbf{R}^{n} \to \mathbf{R}^{m}, then Df(a) exists if all D_{j}f^{i}(x) exists in an open set containing a and if each function D_{j}f^{i} is continuous at a. Such a function f is called continuously differentiable at a.

(Thm 2.9) Let g_{1} , \cdots, g_{m} : \mathbf{R}^{n} \to \mathbf{R} be continuously differentiable at a, and let f : \mathbf{R}^{m} \to \mathbf{R} be differentiable at (g_{1}(a), \cdots, g_{m}(a)). Define the function F : \mathbf{R}^{n} \to \mathbf{R} by F(x) = f(g_{1}(x), \cdots, g_{m}(x)). Then D_{i}F(a) = \sum_{j = 1}^{m} D_{j}f(g_{1}(a), \cdots, g_{m}(a)) \cdot D_{i}g_{j}(a).

(prop) for the function F : \mathbf{R}^{2} \to \mathbf{R} defined by F(x,y) = f(g(x,y), h(x),k(y)) where h,k : \mathbf{R} \to \mathbf{R}. Letting a = (g(x,y), h(x), k(y)), we obtain D_{1}F(x,y) = D_{1}f(a) \cdot D_{1}g(x,y) + D_{2} f(a) \cdot h’(a), D_{2}F(x,y) = D_{1}f(a) \cdot D_{2}g(x,y) + D_{3}f(a) \cdot k’(y).

(subsection) Inverse Functions

(Lem 2.10) Let A \subset \mathbf{R}^{n} be a rectangle and let f : A \to \mathbf{R}^{n} be continuously differentiable . If there is a number M s.t. |D_{j}f^{i}(x) | \le M for all x in the interior of A< then |f(x) – f(y)| \le n^{2} M|x-y| for all x, y \in A.

(Inverse Function Theorem) (Thm 2.110 Suppose that f : \mathbf{R}^{n} \to \mathbf{R}^{n} is continuously differentiable in an open set containing a, and det f’(a) \neq 0. Then there is an open set V containing a and an open set W containing f(a) s.t. f : V \to W has a continuous inverse f^{-1} : W \to V which is differentiable and for all y \in W satisfies (f^{-1})’(y) = [f’(f^{-1}(y))]^{-1}.

an inverse function f^{-1} may exist even if det f’(a) = 0. However, if det f’(a) = 0, then f^{-1} cannot be differentiable at f(a).

(subsection) Implicit Functions

(Implicit Function Theorem) (Thm 2.12) Suppose f : \mathbf{R}^{n} \times \mathbf{R}^{m} \to \mathbf{R}^{m} is continuously differentiable in an open set containing (a,b) and f(a,b) = 0. Let M be the m \times m matrix (D_{n+j}f^{i}(a,b)) 1 \le i,j \le m. If det M \neq 0, there is an open set A \subset \mathbf{R}^{n} containing a and an open set B \subset \mathbf{R}^{m} containing b, with the following property : for each x \in A there is a unique g(x) \in B s.t. f(x,g(x)) = 0. The function g is differentiable. These functions are said to be defined implicitly by the equation f(x,y) = 0.

(Thm 2.13) Let f : \mathbf{R}^{n} \to \mathbf{R}^{p} be continuously differentiable in an open set containing a, where p \le n. If f(a) = 0 and the p \times n matrix(D_{j}f^{i}(a)) has rank p, then there is an open set A \subset \mathbf{R}^{n} containing a and a differentiable function h : A \to \mathbf{R}^{n} with differentiable inverse s.t. f \bullet h (x^{1}, \cdots, x^{n}) = (x^{n-p+1}, \cdots, x^{n}).

Integration

(section 3) Integration

(subsection) Basic definitions

(Def) a partition of a rectangle [a_{1}, b_{1}] \times \cdots \times [a_{n}, b_{n}] is a collection P = (P_{1}, \cdots, P_{n}) , where each P_{i} is a partition of the interval [a_{i},b_{i}].

P = (P_{1}, \cdots, P_{n}) divides [a_{1},b_{1}] \times \cdots \times [a_{n}, b_{n}] into N = N_{1}, \cdots, N_{n} subrectangles. These subrectangles will be called subrectangles of the partition P.

(Def) Supose that A is a rectangle, f : A \to \mathbf{R} is a bounded function, and P is a partition of A. For each subrectangle S of the partition let m_{S}(f) = \inf {f(x) : x \in S} , M_{s}(f) = \sup{f(x) : x \in X}, and let v(S) be the volume of S. [The volume of a rectangle [a_{1}, b_{1}] \times \cdots \times [a_{n}, b_{n}] , and also of (a_{1}, b_{1}) \times \cdots \times (a_{n}, b_{n}) , is defined as (b_{1} – a_{1}) \cdot … \cdot (b_{n} – a_{m}) ] . the lower and upper sums of f for P are defined by

L(f,P) = \sum_{S} m_{S}(f) \cdot v(S) and U(f,P) = \sum_{S} M_{S}(f) \cdot v(S).

(Lem 3.1) Suppose the partition P’ refines P (that is, each subrectangle of P’ is contained in a subrectangle of P). Then L(f,P) \le L(f,P’) and U(f,P’) \le U(f,P) .

(Cor 3.2) If P and P’ are any two partitions, then L(f,P’) \le U(f,P).

The lease upper bound of all lower sums for f is less than or equal to the greatest lower bound of all upper sums for f.

(Def) A function f : A \to \mathbf{R} is called integrable on the rectangle A if f is bounded and \sup{L(f,P)} = \inf{U(f,P)} . This common number is then denoted \int_{A} f, and called the integral of f over A. Often, the notation \int_{A} f(x^{1}, \cdots, x^{n} ) dx^{1} \cdots dx^{n} is used.

(Thm 3-3) A bounded function f : A \to \mathbf{R} is integrable iff for every \epsilon >0 there is a partition P of A s.t. U(f,P) – L(f,P) < \epsilon.

(Prop)

Let f: A \to \mathbf{R} be a constant function, f(x) = c. \int_{A} f = c \cdot v(A).

Let f:[0,1] \times [0,1] \to \mathbf{R} be defined by f(x,y) = \begin{cases} 0 & if x is rational, \ 1 & if x is irrational.\end{cases} . f is not integrable.

(subsection) Measure Zero and Content Zero

(Def) A subset A of \mathbf{R}^{n} has (n-dimensional) measure 0 if for every \epsilon >0 there is a cover {U_{1}, U_{2}, U_{3}, \cdots } of A by closed rectangles s.t. \sum_{i = 1}^{\infty} v(U_{i}) < \epsilon.

A set with only finitely many points has measure 0.

(Thm 3.4) If A = A_{1} \cup A_{2} \cup A_{3} \cdots and each A_{i} has measure 0, then A has measure 0.

(Def) A subset A of \mathbf{R}^{n} has (n-dimensional) content 0 if for every \epsilon>0 there is a finite cover {U_{1}, \cdots, U_{n} } of A by closed rectangles such that \sum_{i = 1}^{n} v(U_{i}) < \epsilon.

(Thm 3.5) If a < b, then [a,b] \subset \mathbf{R} does not have content 0. In fact, if {U_{1}, \cdots, U_{n} } is a finite cover of [a,b] by closed intervals, then \sum_{i = 1}^{n} v(U_{i}) \ge b-a.

(Thm 3.6) If A is compact and has measure 0, then A has content 0.

(subsection) Integrable functions

(Lem 3.7) Let A be a closed rectangle and let f : A \to \mathbf{R} be a bounded function s.t. o(f,x) < \epsilon for all x \in A. Then there is a partition P of A with U(f,P) – L(f,P) < \epsilon \cdot v(A).

(Thm 3.8) Let A be a closed rectangle and f : A \to \mathbf{R} a bounded function. Let B = {x : f is not continuous at x}, then f is integrable iff B is a set of measure 0.

(Def) If C \subset \mathbf{R}^{n}, the characteristic function \chi_{C} of C is defined by \chi_{C}(x) = \begin{cases} 0 & x \notin C, \ 1 & x \in C .\end{cases}

If C \subset A for some closed rectangle A and f : A \to \mathbf{R} is bounded, then \int_{C} f is defined as \int_{A} f \cdot \chi_{C}, provided f \cdot \chi_{C} is integrable.

(Thm 3.9) The function \chi_{C} : A \to \mathbf{R} is integrable iff the boundary of C has measure 0, (and hence content 0).

(Def) A bounded set C whose boundary has measure 0 is called Jordan-measurble. The integral \int_{C} 1 is called (n-dimensional) content of C, or the (n-dimensional) volume of C. one-dimensional is often called length, and two-dimensional volume, area.

(subsection) Fubini’s Theorem

(Def) If f : A \to \mathbf{R} is a bounded function on a closed rectangle, then whether or not f is integrable, the least upper bound of all lower sums, and the greatest lower bound of all upper sums, both exist. They are called the lower and upper integrals of f on A, and denoted \mathbf{L} \int_{A} f and \mathbf{U} \int_{A} f.

(Fubini’s Theorem) (Thm 3.10) Let A \subset \mathbf{R}^{n} and B \subset \mathbf{R}^{m} be closed rectangles, and let f : A \times B \to \mathbf{R} be integrable. for x \in A let g_{x} : B \to \mathbf{R} be defined by g_{x} (y) = f(x,y) and let

\mathcal{L}(x) = \mathbf{L} \int_{B} g_{x} = \mathbf{L} \int_{B} f(x,y) dy,

\mathcal{U}(x) = \mathbf{U} \int_{B} g_{x} = \mathbf{U} \int_{B} f(x,y) dy.

Then \mathcal{L} and \mathcal{U} are integrable on A and

\int_{A \times B} f = \int_{A} \mathcal{L} = \int_{A} (\mathbf{L} \int_{B} f(x,y) dy) dx,

\int_{A \times B} f = \int_{A} \mathcal{U} = \int_{A} (\mathbf{U} \int_{B} f(x,y) dy) dx,

The integrals on the right side are called iterated integrals for f.

(Prop)

\int_{A \times B} f = \int_{A} (\int_{B} f(x,y) dy) dx, if f is continuous.

In the case g_{x} is not integrable for a finite number of x \in A, then \int_{A \times B} f = \int_{A} (\int_{B} f(x,y) dy) dx, provided that \int_{B} f(x,y) dy is defined arbitrarily, say as 0, when it does not exist.

Let f : [0,1] \times [0,1] \to \mathbf{R} be defined by f(x,y) = \begin{cases}1 & if x is irrational \ 1 & if x is rational and y is irrational \ 1 - \frac{1}{q} & if x = \frac{p}{q} in lowest terms and y is rational \end{cases} , Then f is integrable and \int_{[0,1] \times [0,1]} f = 1. Now \int_{0}^{1} f(x,y) dy = 1 if x is irrational, and does not exist if x is rational.

If A = [a_{1}, b_{1} ] \times \cdots \times [a_{n}, b_{n} ] and f : A \to \mathbf{R} is sufficiently nice, \int_{A} f = \int_{a_{n}}^{b_{n}} ( \cdots ( \int_{a_{1}}^{b_{1}} f(x^{1} , \cdots, x^{n}) dx^{1}) \cdots ) dx^{n}.

If C \subset A \times B, Fubini’s theorem can be used to evaluate \int_{C} f, since this is \int_{A \times B} \chi_{C} f.

(subsecton) Partitions of unity

(Thm 3.11) Let A \subset \mathbf{R}^{n} and let \mathcal{O} be an open cover of A. Then there is a collection \Phi| of C^{\infty} functions \phi defined in an open set containing A, with the following properties :

For each x \in A we have 0 \le \phi(x) \le 1.

For each x \in A there is an open set V containing x s.t. all but finitely many \phi \in \Phi are 0 on V.

For each x \in A we have \sum_{\phi \in \Phi} \phi(x) = 1 ( by (2) for each x this sum is finite in some open set containing x)

For each \phi \in \Phi there is an open set U in \mathcal{O} s.t. \phi = 0 outside of some closed set containing in U.

A collection \Phi satisfying 1st and 3rd property is called a C^{\infty} partition of unity for A. If \Phi also satisfies 4th property , it is said to be subordinate to the cover \mathcal{O}.

(prop) Let C \subset A be compact. only finitely many \phi \in \Phi are not 0 on C.

(Def) An open cover \mathcal{O} of an open set A \subset \mathbf{R}^{n} is admissible if each U \in \mathcal{O} is contained in A. If \Phi is subordinate to \mathcal{O}, f : A \to \mathbf{R} is bounded in some open set around each point of A, and {x : f is discontinuous at x} has measure 0, then each \int_{A} \phi \cdot |f| exists. we define f to be integrable (in the extended sense) if \sum_{\phi \in \Phi} \int_{A} \phi \cdot |f| converges.

(Thm 3.12) If \Psi is another partition of unity, subordinate to an admissible cover \mathcal{O}’ of A< then \sum_{\psi \in \Psi} \int_{A} \psi \cdot |f| also converges, and \sum_{\phi \in \Phi} \int_{A} \phi \cdot f = \sum_{\psi \in \Psi} \int_{A} \psi \cdot f.

If A and f are bounded, then f is integrable in the extended sense.

If A is Jordan-measurable and f is bounded, then this definition of \int_{A} f agrees with the old one.

(subsection) Change of Variable)

(Prop) If g : [a,b] \to \mathbf{R} is continuously differentiable and f : \mathbf{R} to \mathbf{R} is continuous, then, \int_{g(a)}^{g(b)} f = \int_{a}^{b} (f \bullet g) \cdot g’ .

(Thm 3.13) Let A \subset \mathbf{R}^{n} be an open set and g : A \to \mathbf{R}^{n} a 1-1, continuously differentiable function s.t. det g’(x) \neq 0 for all x \in A. If f: g(A) \to \mathbf{R} is integrable, then \int_{g(A)} f = \int_{A} (f \bullet g) |det g’ | .

(Sard’s Theorem) (Thm 3.14) Let g : A \to \mathbf{R}^{n} be continuously differentiable, where A \subset \mathbf{R}^{n} is open, and let B = { x \in A : det g’(x) = 0} . Then g(B) has measure 0.

Integration on Chains

(Section 4) Integration on Chains

(subsection) Algebraic Preliminaries

(Def) If V is a vector space (over \mathbf{R}), we will denote the k-fold product V \times \cdots \times V by V^{k} .

A function T : V^{k} \to \mathbf{R} is called multilinear if for each i with 1 \le i \le k we have T(v_{1} , \cdots , v_{i} + v_{i}’ , \cdots , v_{k}) = T(v_{1} , \cdots, v_{i} , \cdots, v_{k} ) + T(v_{1} , \cdots, v_{i}’ , \cdots, v_{k} ) , T(v_{1}, \cdots, av_{i} , \cdots, v_{k}) = aT(v_{1}, \cdots, v_{i} , \cdots, v_{k}) .

(Def) A multilinear function T : V^{k} \to \mathbf{R} is called a k-tensor on V and the set of all k-tensors, denoted \mathcal{J}^{k} (V) and a \in \mathbf{R} we define (S+T) (v_{1}, \cdots, v_{k}) = S(v_{1}, \cdots, v_{k}) + T(v_{1}, \cdots, v_{k}) , (aS)(v_{1}, \cdots, v_{k}) = a \cdot S(v_{1} , \cdots, v_{k} ) .

(Def) If S \in \mathcal{J}^{k} (V) and T \in \mathcal{J}^{l} (V) , we define the tensor product S \otimes T \in \mathcal{J}^{k+1} (V) by S \otimes T (v_{1}, \cdots, v_{k} , v_{k+1}, \cdots, v_{k+l}) = S(v_{1}, \cdots, v_{k}) \cdot T(v_{k+1}, \cdots, v_{k+l}) .

(prop)

(S_{1} + S_{2}) \otimes T = S_{1} \otimes T + S_{2} \otimes T,

S \otimes (T_{1} + T_{2}) = S \otimes T_{1} + S \otimes T_{2},

(aS) \otimes T = S \otimes (aT) = a(S \otimes T),

(S \otimes T) \otimes U = S \otimes (T \otimes U).

(Thm 4.1) Let v_{1} , \cdots, v_{n} be a basis for V, and let \phi_{1} , \cdots, \phi_{n} be the dual basis , \phi_{i} (v_{j}) = \delta_{ij}. Then the set of all k-fold tensor products \phi_{i_{1}} \otimes \cdots \otimes \phi_{i_{k}} 1 \le i_{1}, \cdots, i_{k} \le n is a basis for \mathcal{J}^{k} (V) , which therefore has dimension n^{k}.

(Def) If f : V \to W is a linear transformation, a linear transformation f^{} : \mathcal{J}^{k} (W) \to \mathcal{J}^{k}(V) is defined by f^{}T(v_{1}, \cdots, v_{k}) = T(f(v_{1}) , \cdots, f(v_{k})) for T \in \mathcal{J}^{k}(W) and v_{1}, \cdots, v_{k} \in V. It is easy to verify that f^{} (S \otimes T) = f^{}S \otimes f^{*}T.

(Def) Inner product on V to be a 2-tensor T such that T is symmetric, that is T(v,w) = T(w,v) for v,w \in V and such that T is positive definite, that is, T(v,v) >0 if v \neq 0. We distinguish \langle , \rangle as the usual inner product on \mathbf{R}^{n}.

(Thm 4.2) If T is an inner product on V, there is a basis v_{1}, \cdots, v_{n} for V s.t. T(v_{i}, v_{j}) = \delta_{ij} . (Such a basis is called orthonormal with respct to T.) Consequently there is an isomorphism f : \mathbf{R}^{n} \to V s.t. T(f(x), f(y)) = \langle x, y \rangle for x, y \in \mathbf{R}^{n}. In other words f^{*}T = \langle , \rangle.

(Def) A k-tensor \omega \in \mathcal{J}^{k}(V) is called alternating if \omega(v_{1}, \cdots, v_{i}, \cdots, v_{j}, \cdots, v_{k}) = - \omega(v_{1}, \cdots, v_{j}, \cdots, v_{i}, \cdots, v_{k}) for all v_{1}, \cdots, v_{k} \in V.

The set of all alternating k-tensors is clearly a subspace \Lambda^{k}(V) of \mathcal{J}^{k}(V).

(Def) the sign of a permutation \sigma, denoted \sgn \sigma, is +1 if \sigma is even and -1 if \sigma is odd. If T \in \mathcal{J}^{k}(V), we define Alt(T) by Alt(T)(v_{1}, \cdots, v_{k}) = \frac{1}{k!} \sum_{\sigma \in S_{k}} \sgn \sigma \cdot T(v_{\sigma (1)} , \cdots, v_{\sigma(k)} ), where S_{k} is the set of all permutations of the numbers 1 to k.

(Thm 4.3)

If T \in \mathcal{J}^{k}(V), then Alt (T) \in \Lambda^{k}(V).

If \omega \in \Lambda^{k} (V), then Alt(\omega) = \omega.

If T \in \mathcal{J}^{k} (V), then Alt(Alt(T)) = Alt(T).

(Def) The wedge product \omega \wedge \eta \in \Lambda^{k+l} (V) by \omega \wedge \eta = \frac{(k+l)!}{k!l!} Alt(\omega \otimes \eta).

(prop) (\omega_{1} + \omega_{2}) \wedge \eta = \omega_{1} \wedge \eta + \omega_{2} \wedge \eta,

\omega \wedge (\eta_{1} + \eta_{2} ) = \omega \wedge \eta_{1} + \omega \wedge \eta_{@},

a \omega \wedge \eta = \omega \wedge a \eta = a(\omega \wedge \eta),

\omega \wedge \eta = (-1)^{kl} \eta \wedge \omega,

f^{}(\omega \wedge \eta) = f^{} (\omega) \wedge f^{*} (\eta)

(Thm 4.4)

If S \in \mathcal{J}^{k}(V) and T \in \mathcal{J}^{l}(V) and Alt(S) = 0, then Alt(S \otimes T) = Alt(T \otimes S ) = 0.

Alt(Alt(\omega \otimes \eta) \otimes \theta) = Alt(\omega \otimes \eta \otimes \theta) = Alt(\omega \otimes Alt(\eta \otimes \theta)).

If \omega \in \Lambda^{k}(V), \eta \in \Lambda^{l}(V), and \theta \in \Lambda^{m} (V), then (\omega \wedge \eta) \wedge \theta = \omega \wedge (\eta \wedge \theta) = \frac{(k + l + m)!}{k!l!m!} Alt(\omega \otimes \eta \otimes \theta) .

(Thm 4.5) If v_{1}, \cdots , v_{n} is a basis for V and \phi_{1}, \cdots, \phi_{n} is the dual basis, The set of all \phi_{i_{1}} \wedge \cdots \wedge \phi_{i_{k}} 1 \le i_{1}, < i_{2} < \cdots < i_{k} \le n is a basis for \Lambda^{k} (V), which therefore has dimension \binom{n}{k} = \frac{n!}{k!(n-k)!}.

If V has dimension n, \Lambda^{n} (V) has dimension 1. Thus all alternating n-tensors on V are multiples of any non-zero one.

(Thm 4.6) Let v_{1}, \cdots, v_{n} be a basis for V, and let \omega \in \Lambda^{n} (V). If w_{o} = \sum_{j = 1}^{n} a_{ij}v_{j} are n vectors in V, then \omega(w_{1}, \cdots, w_{n}) = det(a_{ij}) \cdot \omega(v_{1}, \cdots, v_{n}) .

(Def) a nonzero \omega \in \Lambda^{n}(V) splits the bases of V into two disjoint groups, those with \omega(v_{1}, \cdots, v_{n}) > 0 and those for which \omega(v_{1}, \cdots, v_{n}) < 0 ; this criterion is independent of \omega and can always be used to divide the bases of V into two disjoint groups. Either of these two groups is called an orientation for V.

The orientation to which a basis v_{1}, \cdots, v_{n} belongs is denoted [v_{1}, \cdots, v_{n}] and the other orientation is denoted –[v_{1}, \cdots, v_{n} ] . In \mathbf{R}^{n} we define the usual orientation as [e_{1}, \cdots, e_{n} ] .

(Def) For a general vector space V, if an inner product T for V is given. If an orientation \mu for V has also ben given, it follows that there is a unique \omega \in \Lambda^{n} (V) such that \omega(v_{1}, \cdots, v_{n}) = 1 whenever v_{1}, \cdots, v_{n} is an orthonormal basis such that [v_{1}, \cdots, v_{n}] = \mu. This unique \omega is called the volume element of V, determined by the inner product T and orientation \mu.

(Def) If v_{1}, \cdots, v_{n-1} \in \mathbf{R}^{n} and \phi is defined by \phi(w) = det \begin{pmatrix} v_{1} \ \vdots \ v_{n-1} \ w \end{pmatrix}. then \phi \in \Lambda^{1}(\mathbf{R}^{n}) ; therefore there is a unique z \in \mathbf{R}^{n} s.t. \langle w , z \rangle = \phi (w) = det \begin{pmatrix} v_{1} \ \vdots \ v_{n-1} \ w \end{pmatrix} . This z is denoted v_{1} \times \cdots \times v_{n-1} and called the cross product of v_{1}, \cdots, v_{n-1} .

(prop) v_{\sigma(1)} \times \cdots \times v_{\sigma(n-1)} = \sgn \sigma \cdot v_{1} \times \cdots \times v_{n-1},

v_{1} \times \cdots \times av_{i} \times \cdots \times v_{n-1} = a \times (v_{1} \times \cdots \times v_{n-1} ) ,

v_{1} \times \cdots \times (v_{i} + v_{i} ‘) \times \cdots \times v_{n-1} = v_{1} \times \cdots \times v_{i} \times \cdots \times v_{n-1} + v_{1} \times \cdots \times v_{i}’ \times \cdots \times v_{n-1} .

(subsection) Fields and Forms

(Def) If p \in \mathbf{R}^{n} , the sets of all pairs (p,v) , for v \in \mathbf{R}^{n} , is denoted \mathbf{R}^{n}_{p}, and called the tangent space of \mathbf{R}^{n} at p. This set is made into a vector space by (p,v) + (p,w) = (p , v+w) , a \cdot (p,v) = (p, av).

define p + v to be the end point of (p,v). write (p,v) as v_{p} .

(Def) Usual inner product \langle , \rangle_{p} for \mathbf{R}^{n}{p} is defined by \langle v{p} , w_{p} \rangle_{p} = \langle v , w \rangle, and the usual orientation for \mathbf{R}^{n}{p} is [(e{1}){p} , \cdots , (e{n})_{p}] .

(Def) A vector field is a function F s.t. F(p) \in \mathbf{R}^{n}{p} for each p \in \mathbf{R}^{n} . For each p there are numbers F^{1}(p) , \cdots, F^{n} (p) s.t. F(p) = F^{1}(p) \cdot (e{1}){p} + \cdots + F^{n}(p) \cdot (e{n})_{p} . We obtain n component functions F^{i} : \mathbf{R}^{n} \to \mathbf{R}.

Vector field F is called continuous, differentiable, etc., if the functions F^{i} are.

Operations on vectors yield operations on vector fields when applied at each point separately.

(Def) divergence, div F of F, as \sum_{i = 1}^{n} D_{i}F^{i}. \nabla = \sum_{i = 1}^{n} D_{i} \cdot e_{i} , div(F) = \langle \nabla , F \rangle .

(Def) If n = 3 we write (\nabla \times F) (p) = (D_{2} F^{3} – D_{3} F^{2} )(e_{1}){p} + (D{3} F^{1} – D_{1}F^{3})(e_{2}){p} + (D{1} F^{2} – D_{2}F^{1})(e_{3})_{p} . The vector field \nabla \times F is called curl F.

(Def) a function \omega with \omega(p) \in \Lambda^{k}(\mathbf{R}^{n}{p}) . such a function is called a k-form on \mathbf{R}^{n}, or a differential form. If \phi{1}(p) , \cdots, \phi_{n}(p) is the dual basis to (e_{1}){p} , \cdots, (e{n}){p}, then \omega(p) = \sum{i_{1} < \cdots < i_{k}} \omega_{i_{1}, \cdots, i_{k}} (p) \cdot [\phi_{i_{1}} (p) \wedge \cdots \wedge \phi_{i_{k}}(p)] for certain functions \omega_{i_{1}, \cdots, i_{k}} .

The form \omega is called continuous, differentiable, etc., if these functions are.

The sum \omega + \eta, product f \cdot \omega, and wedge product \omega \wedge \eta are defined in the obvious way.

(Def) If f : \mathbf{R}^{n} \to \mathbf{R} is differentiable, then Df(p) \in \Lambda^{1} (\mathbf{R}^{n} . we obtain a 1-form df, defined by df(p)(v_{p}) = Df(p)(v).

(Def) Let x^{i} denote the function \pi_{i}. Since dx^{i}(p)(v_{p}) = d\pi^{i} (p)(v_{p}) = D\pi^{i} (p)(V) = v^{i} , dx^{1}(p), \cdots, dx^{n}(p) is just the dual basis to (e_{1}){p} , \cdots, (e{n}){p} . Thus every k-form \omega can be written \omega = \sum{i_{1} < \cdots < i_{k}} \omega_{i_{1}, \cdots, i_{k}} dx^{i_{1}} \wedge \cdots dx^{i_{k}}

(Thm 4.7) If f : \mathbf{R}^{n} \to \mathbf{R} is differentiable, then df = D_{1}f \cdot dx^{1} + \cdots + D_{n}f \cdot dx^{n}. In classical notation, df = \frac{\partial f}{\partial x^{1}} dx^{1} + \cdots + \frac{\partial f }{\partial x^{n}} dx^{n}.

(Def) Consider now a differentiable function f : \mathbf{R}^{n} \to \mathbf{R} we have a linear transformation Df(p) : \mathbf{R}^{n} \to \mathbf{R}^{m} . we produce linear transformation f_{} : \mathbf{R}^{n}{p} \to \mathbf{R}^{m}{f(p)} defined by f_{}(v_{p}) = (Df(p)(v)){f(p)}. This induces a linear transformation f^{*} : \Lambda^{k}(\mathbf{R}^[m]{f(p)}) \to \Lambda^{k}(\mathbf{R}^{n}_{p}) . If \omega is a k-form on \mathbf{R}^{m} we can therefore define a k-form f^{} \omega on \mathbf{R}^{n} by (f^{} \omega) (p) = f^{*}(\omega(f(p))).

This means that if v_{1}, \cdots, v_{k} \in \mathbf{R}^{n}{p}, then we have f^{*} \omega(p) (v{1}, \cdots, v_{k}) = \omega(f(p))(f_{}(v_{1}, \cdots, f_{}(v_{k})).

(Thm 4.8) If f: \mathbf{R}^{n} \to \mathbf{R}^{m} is differentiable, then

f^{*}(dx^{i}) = \sum_{j = 1}^{n} D_{j}f^{i}\cdot dx^{j} = \sum_{j = 1}^{n} \frac{\partial f^{i}}{\partial x^{j}} dx^{j}.

f^{} (\omega_{1} + \omega_{2}) = f^{}(\omega_{1}) + f^{*} (\omega_{2}) .

f^{} (g \cdot \omega) = (g \bullet f) \cdot f^{} \omega.

f^{} (\omega \wedge \eta) = f^{} \omega \wedge f^{*} \eta.

(Thm 4.9) If f : \mathbf{R}^{n} \to \mathbf{R}^{n} is differentiable, then f^{*} (h dx^{1} \wedge \cdots \wedge dx^{n}) = (h \bullet f)(det f’) dx^{1} \wedge \cdots \wedge dx^{n}.

(Def) If \omega = \sum_{i_{1} < \cdots < i_{k}} \omega_{i_{1}, \cdots, i_{k}} dx^{i_{1}} \wedge \cdots dx^{i_{k}} , we define a (k+1)-form d\omega, the differential of \omega, by d\omega = \sum_{i_{1} < \cdots < i_{k}} d \omega_{i_{1}, \cdots, i_{k}} \wedge dx^{i_{1}} \wedge \cdots dx^{i_{k}} = \sum_{i_{1} < \cdots < i_{k}} \sum_{\alpha = 1}^{n} D_{\alpha}(\omega_{i_{1}, \cdots, i_{k}} ) \cdot dx^{\alpha} \wedge dx^{i_{1}} \wedge \cdots dx^{i_{k}}

(Thm 4.10)

d(\omega + \eta) = d\omega + d\eta.

If \omega is a k-form and \eta is an l-form, then d(\omega \wedge \eta) = d\omega \wedge \eta + (-1)^{k} \omega \wedge d\eta.

d(d\omega)) = 0. Briefly, d^{2} = 0.

If \omega is a k-form on \mathbf{R}^{m} and f : \mathbf{R}^{n} \to \mathbf{R}^{m} is differentiable, then f^{} (d\omega) = d(f^{} \omega).

(Def) a form \omega is called closed if d \omega = 0 and exact if \omega = d \eta, for some \eta.

(Def) An open set A \subset \mathbf{R}^{n} with the properyty that whenever x \in A, the line segment from 0 to x is contained in A; such an open set is called star-shaped with respect to 0.

(Poincare Lemma) (Thm 4.11) If A \subset \mathbf{R}^{n} is an open set star-shaped with respect to 0, then every closed form on A is exact.

(subsection) Geometric preliminaries

(Def) A singular n-cube in A \subset \mathbf{R}^{n} is a continuus function c : [0,1]^{n} \to A (here [0,1]^{n} denotes the n-fold product [0,1] \times \cdots \times [0,1]). \mathbf{R}^{0} and [0,1]^{0} both denote {0}.

A singular 1-cube is often called a curve.

The standard n-cube I^{n} : [0,1]^{n} \to \mathbf{R}^{n} defined by I^{n} (x) = x for x \in [0,1]^{n}.

(Def) a finite sum of singular n-cubes with integer coefficients is called an n-chain in A. For each singular n-chain c in A we shall define an (n-1)-chain in A called the boundary of c and denoted \partial c.

(Def) For each i with 1 \le i \le n we define two singular (n-1)-cubes I^{n}{(i,0)} and I^{n}{(i,1)} as follows. If x \in [0,1]^{n-1}, then

I^{n}_{(i,0)} (x) = I^{n}(x^{1} , \cdots, x^{i-1}, 0, x^{i}, \cdots, x^{n-1}) = (x^{1} , \cdots, x^{i-1}, 0, x^{i}, \cdots, x^{n-1}),

I^{n}_{(i,1)} (x) = I^{n}(x^{1} , \cdots, x^{i-1}, 1, x^{i}, \cdots, x^{n-1}) = (x^{1} , \cdots, x^{i-1}, 1, x^{i}, \cdots, x^{n-1}).

We call I^{n}{(i,0)} the (i,0)-face of I^{n} and I^{n}{(i,1)} the (i,1)-face.

we then define \partial I^{n} = \sum_{i = 1}^{n} \sum_{\alpha = 0,1} (-1)^{i+\alpha} I^{n}_{(i + \alpha)} .

(Def) For a general singular n-cube c: [0,1]^{n} \to A we define the (i,\alpha) -face, c_{(i,\alpha)} = c \bullet (I^{n}{(i,\alpha)}) and then define \partial c = \sum{i = 1}^{n} \sum_{\alpha = 0,1} (-1)^{i+\alpha} c_{(i,\alpha)}.

We define the boundary of an n-chain \sum a_{i}c_{i} by \partial (\sum a_{i} c_{i}) = \sum a_{i} \partial(c_{i}) .

(Thm 4.12) If c is an n-chain in A, then \partial(\partial c) = 0. Briefly, \partial^{2} = 0.

(subsection) The fundamental theorem of calculus

(Def) If \omega is a k-form on [0,1]^{k}, then \omega = f dx^{1} \wedge \cdots \wedge dx^{k} for a unique function f. we define \int_{[0,1]^{k}} \omega = \int_{[0,1]^{k}} f. we could also write this as \int_{[0,1]^{k}} f dx^{1} \wedge \cdots \wedge dx^{k} = \int_{[0,1]^{k}} f(x^{1}, \cdots, x^{k}) dx^{1} \cdots dx^{k} . If \omega is a k-form on A and c is a singular k-cube in A, we define \int_{c} \omega = \int_{[0,1]^{k}} c^{*} \omega.

(Def) If c : {0} \to A is a singular 0-cube in A we define \int_{c} \omega = \omega(c(0)) . The integral of \omega over a k-chain c = \sum a_{i} c_{i} is defined by \int_{c} \omega = \sum a_{i} \int_{c{i}} \omega. The integral of a 1-form over a 1-chain is often called a line integral.

(prop) If P dx + Q dy is a 1-form on \mathbf{R}^{2} and c:[0,1] \to \mathbf{R}^{2} is a singular 1-cube,

\int_{c} P dx + Q dy = \lim \sum_{i = 1}^{n} [c^{1} (t_{i}) – c^{1}(t_{i-1})] \cdot P(c(t^{i})) + [c^{2} (t_{i}) – c^{2}(t_{i-1})] \cdot Q(c(t^{i}))

where t_{0}, \cdots, t_{n} is a partition of [0,1], the choice of t^{i} in [t_{i-1}, t_{i}] is arbitrary, and the limit is taken over all partitions as the maximum of |t_{i} – t_{i-1}| goes to 0.

(Def) Analogous definitions for surface integrals, integrals of 2-forms over singular 2-cubes.

(Stoke’s Theorem) (Thm 4.13) if \omega is a (k-1)-form on an open set A \subset \mathbf{R}^{n} and c is a k-chain in A, then \int_{c} d\omega = \int_{\partial c} \omega.

Integration on Manifolds

(Section 5) Integration on Manifolds

(subsection) Manifolds

(Def) If U and V are open sets in \mathbf{R}{n}, a differentiable function h : U \to V with a differentiable inverse h^{-1} : V \to U will be called a diffeomorphism.

‘Differentiable’ henceforth means C^{\infty}.

(Def) A subset M of \mathbf{R}^{n} is called a k-dimensional manifold (in \mathbf{R}^{n}) if for every point x \in M the following condition is satisfied :

(M) There is an open set U containing x, an open set V \subset \mathbf{R}^{n}, and a diffeomorphism h : U \to V s.t. h(U \cap M) = V \cap (\mathbf{R}^{k} \times {0}) = {y \in V : y^{k+1} = \cdots = y^{n} = 0}.

We say U \cap M is up to diffeomorphism \mathbf{R}^{k} \times {0}.

(Def) n-sphere S^{n}, defined as {x \in \mathbf{R}^{n+1} : |x| = 1}.

(Thm 5.1) Let A \subset \mathbf{R}^{n} be open and let g : A \to \mathbf{R}^{p} be a differentiable function s.t. g’(x) has rank p whenever g(x) = 0. Then g^{-1} (0) is an (n-p)-dimensional manifold in \mathbf{R}^{n}.

(Thm 5.2) A subset M of \mathbf{R}^{n} is a k-dimensional manifold iff for each point x \in M the following coordinate condition is satisfied :

(C) There is an open set U containing x, an open set W \subset \mathbf{R}^{k}, and a 1-1 differentiable function f : W \to \mathbf{R}^{n} s.t. f(W) = M \cap U, f’(y) has a rank k for each y \in W, f^{-1} : f(W) \to W is continuous.

such a function f is called a coordinate system around x.

(Def) The half-space H^{k} \subset \mathbf{R}^{k} is defined as {x \in \mathbf{R}^{k} : x^{k} \ge 0}. A subset M of \mathbf{R}^{n} is a k-dimensional manifold-with-boundary if for every point x \in M either condition (M) or the following condition is satisfied:

(M’) There is an open set U containing x, an open set V \subset \mathbf{R}^{n}, and a diffeomorphism h : U \to V s.t. h(U \cap M) = V \cap (H^{k} \times {0}) = {y \in V : y^{k} \ge 0 and y^{k+1} = \cdots = y^{n} = 0}

and h(x) has kth component 0.

(Def) The set of all points x \in M for which condition M’ is satisfied is called the boundary of M and denoted \partial M.

(subsection) Fields and forms on Manifolds

(Def) Let M be a k-dimensional manifold in \mathbf{R}^{n} and let f : W \to \mathbf{R}^{n} be a coordinate system around x = f(a). Since f’(a) has rank k, the linear transformation f_{} : \mathbf{R}^{k}{a} \to \mathbf{R}^{n}{x} is 1-1, and f_{}(\mathbf{R}^{k}{a}) is a k-dimensional subspace of \mathbf{R}^{n}{x}. If g: V \to \mathbf{R}^{n} is another coordinate system, with x = g(b), then g_{} (\mathbf{R}^{k}{b}) = f{}(f^{-1} \bullet g){*}(\mathbf{R}^{k}{b}) = f_{}(\mathbf{R}^{k}{a}). Thus the k-dimensional subspace f{}(\mathbf{R}^{k}{b}) does not depend on the coordinate system f. This subspace is denoted M{x}, and is called the tangent space of M at x.

a natural inner product T_{x} on M_{x}, induced by that on \mathbf{R}^{n}{x} : if v,w \in M{x} define T_{x}(v,w) = \langle v,w \rangle_{x}.

(prop) Suppose that A is an open set containing M, and F is differentiable vector field on A s.t. F(x) \in M_{x} for each x \in M. If f: W \to \mathbf{R}^{n} is a coordinate system, there is a unique (differentiable) vector field G on W s.t. f_{*}(G(a)) = F(f(a)) for each a \in W.

(Def) Consider a function F which merely assigns a vector F(x) \in M_{x} for each x \in M ; such a function is called a vector field on M.

define F to be differentiable if G corresponding to F is differentiable.

(Def) a function \omega which assigns \omega(x) \in \Lambda^{p}(M_{x}) for each x \in M is called a p-form on M.

If f : W \to \mathbf{R}^{n} is a coordinate system, then f^{} \omega is a p-form on W; we define \omega to be differentiable if f^{} \omega is.

(Thm 5.3) There is a unique (p+1)-form d\omega on M s.t. for every coordinate system f : W \to \mathbf{R}^{n} we have f^{}(d\omega) = d(f^{} \omega).

(Def) when choosing an orientation \mu_{x} for each tangent space M_{x} of a manifold M, such choices are called consistent provided that for every coordinate system f : W \to \mathbf{R}^{n} and a,b \in W the relation [f_{} ((e_{1}){a}), \cdots, f{}((e_{k}){a})] = \mu{f(a)} holds iff [f_{} ((e_{1}){b}), \cdots, f{}((e_{k}){b})] = \mu{f(b)}.

(Def) Suppose orientations \mu_{x} have been chosen consistently. If f: W \to \mathbf{R}^{n} is a coordinate system s.t. [f_{} ((e_{1}){a}), \cdots, f{}((e_{k}){a})] = \mu{f(a)} for one, and hence for every a \in W, then f is called orientation-preserving.

If f is not orientation-preserving and T: \mathbf{R}^{k} \to \mathbf{R}^{k} is a linear transformation with det T = -1, then f \bullet T is orientation-preserving. Therefore there is an orientation-preserving coordinate system around each point.

(Def) A manifold for which orientations \mu_{x} can be chosen consistently is called orientable, and a particular choice of the \mu_{x} is called an orientation \mu of M. A manifold together with an orientation \mu is called an oriented manifold.

(prop) If M is a k-dimensional manifold-with-boundary and x \in \partial M, then (\partial M){x} is a (k-1)-dimensional subspace of the k-dimensional vector space M{x}. Thus there are exactly two unit vectors in M_{x} which are perpendicular to (\partial M)_{x}.

(Def) If f : W \to \mathbf{R}^{n} is a coordinate system with W \subset H^{k} and f(0) = x, then only one of these unit vectors is f_{*}(v_{0}) for some v_{0} with v^{k} <0. This unit vector is called the outward unit normal n(x).

(Def) Suppose that \mu is an orientation of a k-dimensional manifold-with-boundary M. if M is orientable, \parital M is also orientable, and an orientation \mu for M determines an orientation \partial \mu for \partial M, called the induced orientation.

If we apply these definitions to H^{k} with the usual orientation, we find that the induced orientation on \mathbf{R}^{k-1} = {x \in H^{k} : x^{k} = 0} is (-1)^{k} times the usual orientation.

(Def) If M is an oriented (n-1)-dimensional manifold in \mathbf{R}^{n}, a substitute for outward unit normal can be defined. If [v_{1}, \cdots, v_{n-1}] = \mu_{x} , we choose n(x) in \mathbf{R}^{n}{x} so that n(x) is a unit vector perpendicular to M{x} and [n(x), v_{1}, \cdots, v_{n-1}] is the usual orientation of \mathbf{R}^{n}_{x} . We still call n(x) the outward unit normal to M (determined by \mu).

Conversely, if a continuous family of unit normal vectors n(x) is defined on all of M, then we can determine the orientation of M.

(subsection) Stoke’s Theorem on Manifolds

(Def) If \omega is a p-form on a k-dimensional manifold-with-boundary M and c is a singular p-cube in M, we define \int_{c} \omega = \int_{[0,1]^{p}} c^{*} \omega.

In the case p = k there is an open set W \supset [0,1]^{k} and a coordinate system f : W \to \mathbf{R}^{n} s.t. c(x) = f(x) for x \in [0,1]^{k};

if M is oriented, the singular k-cube c is called orientation-preserving if f is.

(Thm 5.4) If c_{1}, c_{2} : [0,1]^{k} \to M are two orientation-preserving singular k-cubes in the oriented k-dimensional manifold M and \omega is a k-form on M s.t. \omega = 0 outside of c_{1}([0,1]^{k}) \cap c_{2}([0,1]^{k}), then \int_{c_{1}} \omega = \int_{c_{2}} \omega.

(Def) Let \omega be a k-form on an oriented k-dimensional manifold M. If there is an orientation-preserving singular k-cube c in M s.t. \omega = 0 outside of c([0,1]^{k}) , we define \int_{m} \omega = \int_{c} \omega.

\int_{M} \omega does not depend on the choice of c.

(prop) Suppose noe that \omega is an arbitrary k-form on M. There is an open cover \mathcal{O} of M s.t. for each U \in \mathcal{O} there is an orientation-preserving singular k-cube c with U \subset c([0,1]^{k}) .

(Def) Let \Phi be a partition of unity for M subordinate to this cover. We define \int_{M} \omega = \sum_{\phi \in \Phi} \int_{m} \phi \cdot \omega provided the sum converges as describes in the discussion preceding Theorem 3-12.

(Def) All our definitions could have been given for a k-dimensional manifold-with-boundary M with orientation \mu.

(Stokes’ Theorem) (Thm 5.5) If M is a compact oriented k-dimensional manifold-with-boundary and \omega is a (k-1)-form on M, then \int_{m} d\omega = \int_{\partial M} \omega. (Here \partial M is given the induced orientation).

(subsection) The Volume element

(Def) Let M be a k-dimensional manifold (or manifold-with-boundary) in \mathbf{R}^{n}, with an orientation \mu. If x \in M, then \mu_{x} and the inner product T_{x} we defined previously determine a volume element \omega(x) \in \Lambda^{k}(M_{x}) . We obtain a nowhere-zero k-form \omega on M, which is called the volume element on M (determined by \mu) and denoted dV.

(Def) the volume of M is defined as \int_{M} dV, provided this integral exists, which is certainly the case if M is compact.

volume is called length or surface area for one-and two-dimensional manifolds, and dV is denoted ds (the element of length) or dA (or dS, the element of surface area).

(Thm 5.6) Let M be an oriented two-dimensional manifold (or manifold-with-boundary) in \mathbf{R}^{3} and let n be the unit outward normal. Then

dA = n^{1} dy \wedge dz + n^{2} dz \wedge dx + n^{3} dx \wedge dy.

Moreover, on M we have

n^{1} dA = dy \wedge dz.

n^{2} dA = dz \wedge dx.

n^{3} dA = dx \wedge dy.

(prop) If c : [0,1] to \mathbf{R}^{n} is differentiable and c([0,1]) is a one-dimensional manifold-with-boundary, the length of c([0,1]) is the least upper bound of the lengths of inscribed broken lines. For c:[0,1]^{2} \to \mathbf{R}^{n}, it is not true.

(subsection) The classical theorems

(Green’s Theorem) (Thm 5.7) Let M \subset \mathbf{R}^{2} be a compact two-dimensional manifold-with-boundary. Suppose that \alpha, \beta : M \to \mathbf{R} are differentiable. Then \int_{\partial M} \alpha dx + \beta dy = \int_{M} (D_{1} \beta – D_{2} \alpha) dx \wedge dy = \int_{M} \int (\frac{\partial \beta}{\partial x}- \frac{\parital \alpha }{\partial y}) dx dy.

(Divergence Theorem) (Thm 5.8) Let M \subset \mathbf{R}^{3} be a compact three-dimensional manifold-with-boundary and n the unit outward normal on \partial M. Let F be a differentiable vector field on M. Then \int_{m} div F dV = \int_{\partial M} \langle F, n \rangle dA.

This equation is also written in terms of three differentiable functions \alpha, \beta, \gamma : M \to \mathbf{R} :

\int \int_{M} \int (\frac{\partial \alpha }{\partial x} + \frac{\partial \beta}{\partial y} +\frac{\partial \gamma}{\partial z}) dV = \int_{\partial M} \int (n^{1} \alpha +n^{2} \beta + n^{3} \gamma) dS.

(Stokes’ Theorem) (Thm 5.9) Let M \subset \mathbf{R}^{3} be a compact oriented two-dimensional manifold-with-boundary and n the unit outward normal on M determined by the orientation of M. Let \partial M have the induced orientation. Let T be the vector field on \partial M with ds(T) = 1 and let F be a differentiable vector field in an open set containing M. Then \int_{M} \langle ( \nabla \times F), n \rangle dA = \int_{\partial M} \langle F, T \rangle ds.

This equation is sometimes written

\int_{\partial M} \alpha dx + \beta dy + \gamma dz = \int_{M} \int [n^{1} (\frac{\partial \gamma }{\partial y } - \frac{\partial \beta }{\partial z}) + n^{2}(\frac{\partial \alpha }{\partial z} - \frac{\partial \gamma }{\partial x }) + n^{3}(\frac{\partial \beta }{\partial x } - \frac{ \partial \alpha }{\partial y}) ] dS.

(Def) If F(x) is the velocity vector of a fluid at x (at some time) then \int_{\partial M} \langle F,n \rangle dA is the amount of fluid ‘diverging’ from M. Consequently the condition div F = 0 expresses the fact that the fluid is incompressible. If M is a disc, the \int_{\partial M} \langle F, T \rangle ds mesures the amount that the fluid curls around the center of the disc. If this is zero for all discs, then \nabla \times F = 0 , and the fluid is called irroational.

Differential Geometry

Curves

(Section 1) Curves

(subsection 1.2.) Parametrized Curves

(Def) R^{3} is denoted the set of triples (x,y,z) of real numbers. A real function of a real variable is differentiable (or smooth) if it has, at all points, derivatives of all orders.

(Def) A parametrized differentiable curve is a differentiable map \alpha : I \to R^{3} of an open interval I = (a,b) of the real line R into R^{3}. The differentiable means that \alpha is a correspondence which maps each t \in I into a point \alpha(t) = (x(t), y(t), z(t)) \in R^{3} which the function x(t), y(t), z(t) are differentiable.

(Def) the vector (x’(t), y’(t), z’(t)) = \alpha ‘(t) \in R^{3} is called the tangent vector (or velocity vector) of the curve \alpha at t. The image set \alpha (I) \subset R^{3} is called the trace of \alpha.

(Def) Let u = (u_{1}, u_{2}, u_{3}) \in R^{3} and define its norms (or length) by |u| = \sqrt{u_{1}^{2} + u_{2}^{2} + u_{3}^{2}}. Let v = (v_{1}, v_{2}, v_{3}) \in R^{3}, and let \theta , 0 \le \theta \le \pi be the angle formed by the segments 0u and 0v. the inner product u \cdot v is defined by u \cdot v = |u||v| cos \theta.

(prop)

Assume that u and v are nonzero vectors. Then u \cdot v = 0 iff u is orthogonal to v.

u \cdot v = v \cdot u.

\lambda(u \cdot v) = \lambda u \cdot v = u \cdot \lambda v.

u \cdot (v + w) = u \cdot v + u \cdot w.

(Def) Let e_{1} = (1,0,0) , e_{2} = (0,1,0) , and e_{3} = (0,0,1) . e_{i} \cdot e_{j} = 1 if i = j and that e_{i} \cdot e_{j} = 1 if i = j and that e_{i} \cdot e_{j} = 0 if i \neq j , where i , j = 1,2,3. Thus, by writing u = u_{1}e_{1} + u_{2}e_{2} + u_{3}e_{3} , v = v_{1}e_{1} + v_{2}e_{2} + v_{3}e_{3} , we obtain u \cdot v = u_{1}v_{1} + u_{2}v_{2} + u_{3}v_{3}.

(subsection 1.3.) Regular Curves ; Arc length

(Def) Let \alpha : I \to R^{3} be a parametrized differentiable curve. For each t \in I where \alpha ‘(t) \neq 0, there is a well-defined straight line which contains the point \alpha (t) and the vector \alpha ‘(t) . This line is called the tangent line to \alpha at t. We call any point where \alpha’(t) = 0 a singular point. We restrict our attention to curves without singular points.

(Def) A parametrized differentiable curve \alpha : I \to R^{3} is said to be regular if \alpha ‘(t) \neq 0 for all t \in I.

Given t \in I, the arc length of a regular parametrized curve \alpha : I \to R^{3} , from the point t_{0}, is by definition s(t) = \int_{t_{0}}^{t} |\alpha ‘(t)| dt, where |\alpha ‘(t)| = \sqrt{(x’(t))^{2} + (y’(t))^{2} + (z’(t))^{2}} is the length of the vector \alpha ‘(t).

(Def) Given the curve \alpha parametrized by arc length s \in (a,b) , we may consider the curve \beta defined in (-b, -a) by \beta(-s) = \alpha (s) , which has the same trace as the first one but is described in the opposite direction. We say that these two curves differ by a change of orientation.

(subsection 1.4.) The Vector product in R^{3}

(Def) Two ordered basis e = {e_{i}} and f = {f_{i}} , i = 1,…,n, of an n-dimensional vector space V has the same orientation if the matrix of change of basis has positive determinant. We denote this relation by e ~ f.

e~f is an equivalence relation. Each of equivalence classes determined by the above relation is called an orientation of V. In the case V = R^{3}, there exists a natural ordered basis e_{1} = (1,0,0) , e_{2} = (0,1,0) , and e_{3} = (0,0,1). We call the orientation corresponding the positive orientation of R^{3}, the other one being the negative orientation.

(Def) Let u, v \in R^{3} . The vector product of u and b is the unique vector u \wedge v \in R^{3} characterized by (u \wedge v) \cdot w = det (u,v,w) for all w \in R^{3}. Here det (u,v,w) means that if we express u,v, and w in the natural basis {e_{i}} , u = \sum u_{i}e_{i} , v = \sum v_{i}e_{i} , w = \sum w_{i}e_{i} , i = 1,2,3, then det(u,v) = \begin{vmatrix} u_{1}& u_{2} &u_{3} \ v_{1}& v_{2} &v_{3} \ w_{1} &w_{2} &w_{3} \end{vmatrix}. Usually u \wedge v is written u \times v and denoted as cross product.

(prop) u \wedge v = - v \wedge u.

u \wedge v depends linearly on u and v; for any real numbers a,b, we have (au + bw) \wedge v = au \wedge v + bw \wedge v.

u \wedge v = 0 iff u and v are linearly dependent.

(u \wedge v) \cdot u = 0, (u \wedge v) \cdot v = 0.

The vector product of u and v is a vector u \wedge v perpendicular to a plane generated by u and v, with a norm equal to the area of a parallelogram generated by u and v and a direction s.t. (u,v, \u \wedge v) is a positive basis.

(subsection 1.5.) The Local Theory of Curves Parametrized by Arc Length

(Def) Let \alpha : I \to R^{3} be a curve parametrized by arc length s \in I. The number |\alpha ‘’(s) | = k(s) is called the curvature of \alpha at s.

(Def) At points where k(s) \neq 0, a unit vector n(s) in the direction \alpha’’(s) is well defined by the equation \alpha’’(s) = k(s) n(s). n(s) is normal to \alpha’(s) and is called the normal vector at s. The plane determined by the unit tangent and normal vectors, \alpha’(s) and n(s), is called the osculating plane at s.

s \in I is a singular point of order 1 if \alpha’’(s) = 0. We denote t(s) = \alpha’(s) the unit tangent vector of \alpha at s. Thus, t’(s) = k(s)n(s).

The unit vector b(s) = t(s) \wedge n(s) is normal to the osculating plane and will be called the binormal vector at s. It measures how rapidly the curve pulls away from the osculating plane at s , in a neighborhood of s. b’(s) is parallel to n(s), and we may write b’(s) = \tau(s) n(s) for some function \tau (s).

(Def) Let \alpha : I \to R^{3} be a curve parametrized by arc length s s.t. \alpha’’(s) \neq 0 , s \in I. The number \tau(s) defined by b’(s) = \tau(s) n(s) is called the torsion of \alpha at s.

(Def) To each value of the parameter s, we have associated three orthogonal unit vectors t(s), n(s), b(s) . The trihedrom formed is referred to as the Frenet trihedron at s. We call the equations t’ = kn , n’ = -kt - \tau b, b’ = \tau n the Frenet formulas. The tb plane is called the rectifying plane, and the nb plane the normal plane. The lines which contain n(s) and b(s) and pass through \alpha(s) are called the principal normal and the binormal, respectively. The inverse R = 1/k of the curvature is called the radius of curvature at s.

(Fundamental Theorem of the Local Theory of Curves) Given differentiable functions k(s) >0 and \tau(s), s \in I, there exists a regular parametrized curve \alpha : I \to R^{3} s.t. s is the arc length, k(s) is the curvature, and \tau(s) is the torsion of \alpha. Moreover, any other curve \bar{\alpha} , satisfying the same conditions, differs from \alpha by a rigid motion ; that is , there exists an orthogonal linear map \rho of R^{3}, with positive determinant, and a vector c such that \bar{\alpha} = \rho \bullet \alpha + c.

(subsection 1.6.) The Local canonical form

(Def) Let \alpha : I \to R^{3} be a curve parametrized by arc length without singular points of order 1. We write the equations of the curve, in a neighborhood of s_{0}, using the trihedron t(s_{0}) , n(s_{0}), b(s_{0}) as a basis for R^{3}. WLOG , s_{0} = 0. Consider the taylor expansion \alpha(s) = \alpha(0) + s(\alpha’(0)) + \frac{s^{2}}{2} \alpha’’(0) + \frac{s^{3}}{6} \alpha’’’(0) + R where \lim_{s \to 0} R/s^{3} = 0. Let us now take the system Oxyz in such a way that the origin O agrees with \alpha(0) and that t = (1,0,0) , n = (0,1,0) , b = (0,0,1). Under these conditions, \alpha(s) = (x(s), y(s), z(s)) is given by x(s) = s - \frac{k^{2}s^{2}}{6} + R_{x} , y(s) = \frac{k}{2} s^{2} + \frac{k’s^{3}}{6} + R_{y} , z(s) = -\frac{k \tau}{6} s^{3} + R_{z} ,where R = (R_{x} , R_{y}, R_{z}) . This representation is called the local canonical form of \alpha, in a neighborhood of s = 0.

(prop) Existence of a neighborhood J \subset I of s = 0 s.t. \alpha(J) is entirely contained in the one side of the rectifying plane toward which the vector n is pointing.

The osculating plane at s is the limit position of the plane determined by the tangent line at s and the point \alpha(s + h) when h \to 0.

(subsection 1.7.) Global Properties of Plane Curves

(Def) A differentiable function on a closed interval [a,b] is the restriction of a differentiable function defined on an open interval containing [a,b].

A closed plane curve is a regular parametrized curve \alpha : [a,b] \to R^{2} s.t. \alpha and all its derivatives agree at a and b; that is, \alpha (a) = \alpha (b), \alpha’(a) = \alpha’(b) , …

The curve \alpha is simple if it has no further self-intersections.

We usually consider the curve \alpha : [0, l] \to R^{2} parametrized by arc length s; hence, l is the length of \alpha.

We assume that a simple closed curve C in the plane bounds a region of this plane that is called the interior of C. We assume that the parameter of a simple closed curve can be so chosen that if one is going along the curve in the direction of increasing parameters, the interior of the curve remains left. Such a curve will be called positively oriented.

(subsubsection A) The Isoperimetric Inequality.

(Def) The area A bounded by a positively oriented simple closed curve \alpha (t) = (x(t), y(t)), where t \in [a,b] is an arbitrary parameter : A = -\int_{a}^{b} y(t) x’(t) dt = \int_{a}^{b} x(t) y’(t) dt = \frac{1}{2} \int_{a}^{b} (xy’ – yx’) dt.

(The Isoperimetric Inequality) (Thm 1) Let C be a simple closed plane curve with length l, and let A be the area of the region bounded by C. Then l^{2} – 4 \pi A \ge 0 , and equality holds iff C is a circle.

This applies to C^{1} curves, that is, curves \alpha(t) = (x(t), y(t)) , t \in [a,b] for which we require only that the functions x(t), y(t) have continuous first derivatives. It holds for piecewise C^{1} curves, which is continuous curves that are made up by a finite number of C^{1} arcs.

(subsubsection B) The Four-Vertex Theorem

(Def) Let \alpha : [0,l] \to R^{2} be a plane closed curve given by \alpha(s) = (x(s), y(s)) . Since s is the arc length, the tangent vector t(s) = (x’(s), y’(s)) has unit length. The tangent indicatrix t : [0,l] \to R^{2} that is given by t(s) = (x’(s), y’(s)) is a differentiable curve, the trace of which is contained in a circle of radius 1.

Define a global differentiable function \theta : [0,l] \to R by \theta(s) = \int_{0}^{s} k(s) ds . Since \alpha is closed, this angle is an integer multiple I of 2 \pi ; that is , \int_{0}^{l} k(s) ds = \theta(l) - \theta (0) = 2 \pi I. This integer I is called the rotation index of the curve \alpha.

(prop)(Theorem of Turning Tangents) The rotation index of a simple closed curve is \mp 1, where the sign depends on the orientation of the curve.

(Def) A regular, plane curve \alpha : [a,b] \to R^{2} is convex if, for all t \in [a,b] , the trace \alpha([a,b]) of \alpha lies entirely on one side of the closed half-plane determined by the tangent line at t.

A vertex of a regular plane curve \alpha : [a,b] \to R^{2} is a point t \in [a,b] where k’(t) = 0.

(The Four-vertex Theorem) (Thm 2) A simple closed convex curve has at least four vertices.

(Lemma) Let \alpha : [0,1] \to R^{2} be a plane closed curve parametrized by arc length and let A,B, and C be arbitrary real numbers. Then \int_{0}^{1} (Ax + By + C) \frac{dk}{ds} ds = 0 where the functions x = x(s), y = y(s) are given by \alpha(s) = (x(s), y(s)) , and k is the curvature of \alpha.

(subsubsection C) The Cauchy-Crofton Formula

(Def) Let C be a regular curve in the plane. We look at all straight lines in the plane that meet C and assign to each such line a multiplicity which is the number of its intersection points with C.

A straight line in the plane can be thought as a point in a plane given by two parameters \rho and \theta which determines a line. Define a measure (area) of a subset of straight lines in the plane, by the area of certain plane.

(The Cauchy-Crofton Formula) (Thm 3) Let C be a regular plane curve with length l. The measure of the set of straight lines (counted with multiplicities) which meet C is equal to 2l.

(Def) A rigid motion in R^{2} is a map F : R^{2} \to R^{2} given by (\bar{x}, \bar{y}) \to (x,y) where x = a + \bar{x} cos \phi - \bar{y} sin \phi , y = b + \bar{x} sin \phi + \bar{y} cos \phi.

(Prop 1) Let f (x,y) be a continuous function defined in R^{2}. For any set S \subset R^{2}, define the area A of S by A(S) = \int \int_{S} f(x,y) dx dy (on only those sets for which the abouve integral exists). Assume that A is invariant under rigid motions; that is , if S is any set and \bar{S} = F^{-1}(S), where F is the rigid motion, we have A(\bar{S}) = \int \int_{S} f(\bar{x} , \bar{y}) d \bar{x} d \bar{y} = \int \int_{S} f(x,y) dx dy = A(S). Then f(x,y) = const.

Jacobian of a rigid motion is 1, and the rigid motions are transitive on points of the plane; that is, given two points in the plane there exists a rigid motion taking one point into the other.

(Def) In the set of all straight lines in the plane \mathcal{L} = {(p, \theta) \in R^{2} ; (p,\theta) ~ (p , \theta + 2k\pi) and (p, \theta) ~ (-p, \theta \mp \pi)}, from the prop 1, define the measure of a set \mathcal{S} \subset \mathcal{L} as \int \int_{\mathcal{S}} d \rho d \theta in the same way as prop 1.

(App) If a curve is not rectifiable but the \int \int n d \rho d \theta (Let n = n(\rho, \theta) be the number of intersection points of the straight line (\rho , \theta)) has a meaning, this can be used to determine the length of such a curve.

Consider a family of parallel straight lines s.t. two consecutive lines are at a distance r. Rotate this family by angles of \pi/4, 2 \pi/4. 3 \pi/4 in order to obtain four families of straight lines. Let n be the number of intersection points of a curve with all these lines. Then \frac{1}{2} nr \frac{\pi}{4} is an approximation to length of C.

Regular Surfaces

Section 2) Regular Surfaces

(subsection 2.2) Regular Surfaces : Inverse Images of Regular Values

(Def 1) A subset S \subset R^{3} is a regular surface if, for each p \in S, there exists a neighborhood V in R^{3} and a map \mathbf{x} : U \to V \cap S of an open set U \subset R^{3} s.t.

\mathbf{x} is differentiable. This means that if we write \mathbf{x} (u,v) = (x(u,v), y(u,v), z(u,v)) , (u, v) \in U, the functions x(u,v) , y(u,v), z(u,v) have continuous partial derivatives of all orders in U.

\mathbf(x) is homeomorphism ; \mathbf{x} has an inverse \mathbf{x}^{-1} : V \cap S \to U which is continuous ; that is , \mathbf{x}^{-1} is the restriction of a continuous map F : W \subset R^{3} \to R^{2} defined on an open set W containing V \cap S.

(The regularity condition) For each q \in U, the differential d\mathbf{x}{q} : R^{2} \to R^{3} is one-to-one; d\mathbf{x}{q} = \begin{pmatrix} \frac{\partial x}{\partial u} &\frac{\partial x}{\partial v} \ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} \ \frac{\partial z}{\partial u}& \frac{\partial z}{\partial v} \end{pmatrix} \right|{q = (u{0}, v_{0}) each column is linearly independent.

This mapping \mathbf{x} is called a parametrization or a system of (local) coordinates in (a neighborhood of) p. The neighborhood V \cap S of p in S is called a coordinate neighborhood.

(Prop 1) If f : U \to R is a differentiable function in an open set U of R^{2}. then the graph of f, that is , the subset of R^{3} given by (x,y,f(x,y)) for (x,y) \in U, is a regular surface.

(Def 2) Given a differential map F : U \subset R^{n} \to R^{m} defined in an open set U of R^{n} we say that p \in U is a critical point of F if the differential dF_{p} : R^{n} \to R^{m} is not a surjective mapping. The image F(p) \in R^{m} of a critical point is called a critical value of F. A point of R^{m} which is not a critical value is called a regular value of F.

(prop 2) If f : U \subset R^{3} \to R is a differentiable function and a \in f(U) is a regular value of f, then f^{-1}(a) is a regular surface in R^{3}.

(prop 3) Let S \subset R^{3} be a regular surface and p \in S. Then there exists a neighborhood V of p in S s.t. V is the graph of a differentiable function which has one of the following three forms : z = f(x,y) , y = g(x,z), x = h(y,z).

(prop 4) Let p \in S be a point of a regular surface S and let \mathbf{x} : U \subset R^{2} \to R^{3} be a map with p \in \mathbf{x}(U) \subset S s.t. conditions 1 and 3 of (Def 1) holds. Assume that \mathbf{x} is one-to-one. then \mathbf{x}^{-1} is continuous.

(subsection 2.3.) Change of Parameters : Differentiable Function on Surfaces

(Change of Parameters)(Prop 1) Let p be a point of a regular surface S , and let \mathbf{X} : U \subset R^{2} \to S, y : V \subset R^{2} \to S be two parametrization of S s.t. p \in \mathbf{x}(U) \cap y(V) = W. Then the “Change of coordinates” h = \mathbf{x}^{-1} \bullet y : y^{-1}(W) \to \mathbf{x}^{-1}(W) is a diffeomorphism ; that is , h is differentiable and has a differentiable inverse h^{-1}.

(Def 1) Let f : V \subset S \to R be a function defined in an open subset V of a regular surface S. Then f is said to be differentiable at p \in V if, for some parametrization x : U \subset R^{2} \to S with p \in x(U) \subset V, the composition f \bullet x : U \subset R^{2} \to R is differentiable at x^{-1}(p) . f is differentiable in V if it is differentiable at all points of V.

(Def) Two regular surfaces S_{1}, S_{2} are diffeomorphic if there exists a differentiable map \phi : S_{1} \to S_{2} with a differentiable inverse \phi^{-1} : S_{2} \to S_{1} . such a \phi is called a diffeomorphism from S_{1} to S_{2}.

(Def 2) A parametrized surface x : U \subset R^{2} \to R^{3} is a differentiable map x from an open set U \subset R^{2} into R^{3}. The set x(U) \subset R^{3} is called the trace of x. x is regular if the differential dx_{q} : R^{2} \to R^{3} is one-to-one for all q \in U (the vectors \partial x / \partial u , \partial x / \partial v are linearly independent for all q \in U) . A point p \in U where dx_{q} is not one-to-one is called a singular point of x.

(Prop 2) Let x : U \subset R^{2} \to R^{3} be a regular parametrized surface and let q \in U. Then there exists a neighborhood V of q in R^{2} s.t. x(V) \subset R^{3} is a regular surface.

(subsection 2.4.) The tangent plane ; The differential of a map

(prop 1) Let x : U \subset R^{2} \to S be a parametrization of a regular surface S and let q \in U. The vector subspace of dimension 2, dx_{q} (R^{2}) \subset R^{3} coincides with the set of tangent vectors to S at x(q).

The plane dx_{q}(r^{2}) , which passes through x(q) = p , which does not depend on the parametrization x, will be called the tangent plane to S at p and will be denoted T_{p}(S).

(Def) The coordinates of a vector w \in T_{p}(S) in the basis associated to a parametrization x are determined as follows. w is the velocity vector \alpha’(0) of a curve \alpha = x \bullet \beta, where \beta : (-\epsilon, \epsilon) \to U is given by \beta(t) = (u(t), v(t)) , with \beta(0) = q = x^{-1}(p).

(prop2) In the discussion above, given w, the vector \beta ‘(0) does not depend on the choice of \alpha. The map d\phi_{p} : T_{p}(S_{1}) \to T_{\phi(p)}(S_{2}) defined by d\phi_{p}(w) = \beta’(0) is linear.

(Def) The linear map d\phi_{p} defined by (prop2) is called the differential of \phi at p \in S_{1}. In a similar way we define the differential of a (differentiable) function f : U \subset S \to R at p \in U as a linear map df_{p} : T_{p}(S) \to R.

A mapping \phi : U \subset S_{1} \to S_{2} is a local diffeomorphism at p \in U if there exists a neighborhood V \subset U of p s.t. \phi restricted to V is a diffeomorphism onto an open set \phi(V) \subset S_{2}.

(prop 3) If S_{1} and S_{2} are regular surfaces and \phi : U \subset S_{1} \to S_{2} is a differentiable mapping of an open set U \subset S_{1} s.t. the differential d\phi_{p} of \phi at p \in U is an isomorphism, then \phi is a local diffeomorphism at p.

(subsection 2.5.) The First Fundamental form ; Area

(Def) The natural Inner product of R^{3} \supset S induces on each tangent plane T_{p}(S) of a regular surface S an inner product, to be denoted by < , >{p} : If w{1}, w_{2} \in T_{p} (S) \subset R^{3} , then <w_{1}, w_{2}>{p} is equal to the inner product of w{1} and w_{2} as vectors in R^{3}. There corresponds a quadratic form I_{p} : T_{p}(S) \to R given by I_{p}(w) = <w,w>_{p} = |w|^{2} \ge 0.

(Def 1) The quadratic form I_{p} on T_{q}(S), defined as above, is called the first fundamental form of the regular surface S \subset R^{3} at p \in S.

(Def) For a regular surface S, a (regular) domain of S is an open and connected subset of S s.t. its boundary is the image of a circle by a differentiable homeomorphism which is regular (that is, its differential is nonzero) except at a finite number of points. A region of S is the union of a domain with its boundary. A region of S \subset R^{3} is bounded if it is contained in some ball of R^{3}.

(Def 2) Let R \subset S be a bounded region of a regular surface contained in the coordinate neighborhood of the parametrization x : U \subset R^{2} \to S. The positive number \int \int_{Q} |x_{u} \wedge x_{v}| du dv = A(R), Q = x^{-1}(R) is called the area of R.

(subsection 2.6.) Orientation of Surfaces

(Def 1) A regular surface S is called orientable if it is possible to cover it with a family of coordinate neighborhoods in such a way that if a point p \in S belongs to two neighborhoods of this family, then the change of coordinates has positive Jacobian at p. The choice of such a family is called an orientation of S, and S, in this case, is called oriented. If such a choice is not possible, the surface is called nonorientable.

If a regular surface can be covered by two coordinate neighborhoods whose intersection is connected, then the surface is orentable.

(Prop1) A regular surface S \subset R^{3} is orientable iff there exists a differentiable field of unit normal vectors N : S \to R^{3} on S.

(prop 2) If a regular surface is given by S = {(x,y,z) \in R^{3} : f(x,y,z) = a}, where f : U \subset R^{3} \to R is differentiable and a is a regular value of f, then S is orientable.

(subsection 2.7.) A Characterization of Compact Orientable Surfaces

(Def) Let S \subset R^{3} be an orientable surface. On the normal line through p \in S, an open interval I_{p} around p of length, say, 2 \epsilon_{p} (\epsilon_{p} varies with p) in such a way that if p \neq q \in S, then I_{p} \cap I_{q} = \empty. Thus, the union \bigcup I_{p} , p \in S, constitutes an open set V of R^{3}, which contains S and has the property that through each point of V there passes a unique normal line to S; V is then called a tubular neighborhood of S.

(Prop 1) Let S be a regular surface and x : U \to S be a parametrization of a neighborhood of a point p = x(u_{0}, v_{0}) \in S. Then there exists a neighborhood W \subset x(U) of p in S and a number \epsilon >0 s.t. the segments of the normal lines passing through points q \in W, with center at q and length 2 \epsilon, are disjoint. (That is, W has a tubular neighborhood.)

(prop 2) Assume the existence of a tubular neighborhood V \subset R^{3} of an orientable surface S \subset R^{3}, and choose an orientation for S. Then the function g : V \to R , defined as the oriented distance form a point of V to the foot of the unique normal line passing through this point, is differentiable and has zero as a regular value.

(Bolzano-Weierstrass) (Prop 1) Let A \subset R^{3} be a compact set. Then every infinite subset of A has at least one limit point in A.

(Heine-Borel) (Prop 2) Let A \subset R^{3} be a compact set and {U_{\alpha}} be a family of open sets of A s.t. \bigcup_{\alpha} U_{\alpha} = A. Then it is possible to choose a finite number U_{k_{1}}, U_{k_{2}}, …, U_{k_{n}} of U_{\alpha} s.t. \bigcup U_{k_{i}} = A , i = 1,…,n.

(Lebesque) (prop 3) Let A \subset R^{3} be a compact set and {U_{\alpha}} be a family of open sets of A s.t. \bigcup_{\alpha} U_{\alpha} = A. Then there exists a number \delta >0 (The Lebesgue number of the family {U_{\alpha}}) s.t. whenever two points p,q \in A are at a distance d(p,q) < \delta then p and q belong to some U_{\alpha}.

(Prop 3) Let S \subset R^{3} be a regular, compact, orientable surface. Then there exists a number \epsilon >0 s.t. whenever p, q \in S the segments of the normal lines of length 2 \epsilon , centered in p and q, are disjoint( that is, S has a tubular neighborhood).

(Thm) Let S \subset R^{3} be a regular compact orientable surface. Then there exists a differentiable function g : V \to R , defined in an open set V \subset R^{3}, with V \supset S (precisely a tubular neighborhood of S), which has zero as a regular value and is such that S = g^{-1}(0).

(subsection 2.8.) A geometric definition of Area

(Def) For a region R \subset S we define a partition \mathcal{P} of R into a finite number of regions R_{i}, that is, we write R = \bigcup_{i} R_{i} , where intersection of two such regions R_{i} is either empty or made up of boundary points of both regions. The diameter of R_{i} is the supremum of the distances (in R^{3}) of any two points in R_{i}. The largest diameter of the R_{i}’s of a given partition \mathcal{P} is called the norm \mu of \mathcal{P}. If we take a partition of each R_{i}, we obtain a second partition of R, which is said to refine \mathcal{P}.

Given a partition R = \bigcup_{i} R_{i} of R, we choose arbitrarily points p_{i} \in R_{i} and project R_{i} onto the tangent palne at p_{i} in the direction of normal line at p_{i}; this projection is denoted by \bar{R} and its area b A(\bar{R}) .

If, by choosing partitions \mathcal{P}{1}, …, \mathcal{P}{n}, … more and more refined s.t. the norm \mu_{n} of \mathcal{P}{n} converges to zero, there exists a limit of \sum{i} A(\bar{R}{i}) and this limit is independent of all choices. R has an area A(R) defined by A(R) = \lim{\mu_{n} \to 0} \sum_{i} A(\bar{R_{i}}).

(Prop 1) Let x : U \to S be a coordinate system in a regular surface S and let R = x(Q) be a bounded region of S contained in x(U). Then R has an area given by A(R) = \int \int_{Q} |x_{u} \wedge x_{v}| du dv.

(Appendix) A brief review of continuity and Differentiability

(Prop 1) F : U \subset R^{n} \to R^{m} is continuous iff each component function f_{i} : U \subset R^{n} \to R , i = 1,…,m, is continuous.

(Prop 2) A map F : U \subset R^{n} \to R^{m} is continuous at p \in U iff , given a neighborhood V of F(p) in R^{m} there exists a neighborhood W of p in R^{n} s.t. F(W) \subset V.

(Prop 3) Let F : U \subset R^{n} \to R^{m} and G : V \subset R^{m} \to R^{k} be continuous maps, where U and V are open sets s.t. F(U) \subset V. Then G \bullet F : U \subset R^{n} \to R^{k} is a continuous map.

(Intermediate Value Theorem) (Prop 4) Let f : [a,b] \to R be a continuous function defined on the closed interval [a,b] . Assume that f(a) and f(b) have opposite signs; that is, f(a)f(b) < 0. Then there exists a point c \in (a,b) s.t. f(c) = 0.

(prop 5) Let f : [a,b] be a continuous function defined in the closed interval [a,b]. Then f reaches its maximum and its minimum in [a,b]; that is, there exists points x_{1}, x_{2} \in [a,b] s.t. f(x_{1}) \le f(x) \le f(x_{2}) for all x \in [a,b].

(Heine-Borel) (prop 6) Let [a,b] be a closed interval and let I_{\alpha} , \alpha \in A be a collection of open intervals in [a,b] s.t. \bigcup_{\alpha} I_{\alpha} = [a,b] . Then it is possible to choose a finite number I_{k_{1}}, I_{k_{2}}, …, I_{k_{n}} of I_{\alpha} s.t. \bigcup I_{k_{i}} = I, i = 1,…,n.

(subsection B) Differentiability in R^{n}

(Def 1) Let F : U \subset R^{n} \to R^{m} be a differentiable map. To each p \in U we associate a linear map dF_{p} : R^{n} \to R^{m} which is called the differential of F at p and is defined as follows. Let w \in R^{n} and let \alpha : (- \epsilon, \epsilon) \to U be a differentiable curve s.t. \alpha(0) = p, \alpha’(0) = w. By the chain rule, the curve \beta = F \bullet \alpha : (-\epsilon , \epsilon) \to R^{m} is also differentiable. Then dF_{p}(w) = \beta’(0) .

(prop 7.) The above definition of dF_{p} does not depend on the choice of the curve which passes through p with tangent vector w, and dF_{p} is , in fact, a linear map.

(The Chain rule for Maps) (Prop 8) Let F : U \subset R^{n} \to R^{m} and G : V \subset R^{m} \to R^{k} be differentiable maps, where U and V are open sets s.t. F(U) \subset V. Then G \bullet F : U \to R^{k} is a differentiable map, and d(G \bullet F){p} = dG{F(p)} \bullet dF_{p} , p \in U.

(Prop 9) Let f : U \subset R^{n} \to R be a differentiable function defined on a connected open subset U of R^{n}. Assume that df_{p} : R^{n} \to R is zero at every point p \in U. Then f is constant on U.

(Inverse Function Theorem) Let F : U \subset R^{n} \to R^{n} be a differentiable mapping and suppose that at p \in U the differential dF_{p} : R^{n} \to R^{n} is an isomorphism. Then there exists a neighborhood V of p in U and a neighborhood W of F(p) in R^{n} s.t. F:V \to W has a differentiable inverse F^{-1} : W \to V.

(Def) A differentiable mapping F : V \subset R^{n} \to W \subset R^{n}, where V and W are open sets, is called a diffeomorphism of V with W if F has a differentiable inverse.

The Geometry of the Gauss Map

(section 3) The Geometry of the Gauss Map

(subsection 3.2.) The Definition of the Gauss Map and its Fundamental properties

(Def) If V \subset S is an open set in S and N : V \to R^{3} is a differentiable map which associates to each q \in V a unit normal vector at q , we say that N is a differentiable field of unit normal vectors on V.

A regular surface is orientable if it admits a differentiable field of unit normal vectors defined on the whole surface; the choice of such a field N is called an orientation of S.

(Def 1) Let S \subset R^{3} be a surface with an orientation N. The map N : S \to R^{3} takes its values in the unit sphere S^{2} = {(x,y,z) \in R^{3} ; | x^{2} + y^{2} + z^{2} = 1} The map N : S \to S^{2} , thus defined, is called the Gauss map of S.

(Prop 1) The differential dN_{p} : T_{p}(S) \to T_{p} (S) of the Gauss map is a self-adjoint linear map.

(Def 2) The quadratic form II_{p}, defined in T_{p}(S) by II_{p}(v) = - \langle dN_{p}(v), v \rangle is called the second fundamental form of S at p.

(Def 3) Let C be a regular curve in S passing through p \in S, k the curvature of C at p, and cos \theat = \langle n, N \rangle, where n is the normal vector to C and N is the normal vector to S at p. The number k_{n} = k cos \theta is then called the normal curvature of C \subset S at p.

(Meusnier) (prop 2) All curves lying on a surface S and having at a given point p \in S the same tangent line have at this point the same normal curvatures.

(Def 4) The maximum normal curvature k_{1} and the minimum normal curvature k_{2} are called the principal curvatures at p; the corresponding directions, that is, the directions given by the eigenvectors e_{1}, e_{2} are called principal directions at p.

(Def 5) If a regular connected curve C on S is such that for all p \in C the tangent line of C is a principal direction at p, then C is said to be a line of curvature of S.

(Olinde Rodrigues) (Prop 3) A necessary and sufficient condition for a connected regular curve C on S to be a line of curvature of S is that N’(t) = \lambda(t) \alpha’(t) for any parametrization \alpha(t) of C, where N(t) = N \bullet \alpha(t) and \lambda (t) is a differentiable function on t. In this case, \lambda(t) is the (principal) curvature along \alpha’(t).

(Def 6) Let p \in S and let dN_{p} : T_{p}(S) \to T_{p}(S) be the differential of the Gauss map. The determinant of dN_{p} is the Gaussian curvature K of S at p. The negative of half of the trace of dN_{p} is called the mean curvature H of S at p.

(Def 7) A point of a surface S is called

Elliptic if det(dN_{p}) >0.

Hyperbolic if det(dN_{p}) < 0.

Parabolic if det(dN_{p}) = 0 , with dN_{p} \neq 0.

Planar if dN_{p} = 0.

(Def 8) If at p \in S, k_{1} = k_{2}, then p is called an umbilical point of S; in particular, the planar points (k_{1} = k_{2} = 0) are umbilical points.

(Prop 4) If all points of a connected surface S are umbilical points, then S is either contained in a sphere or in a plane.

(Def 9) Let p be a point in S. An asymptotic direction of S at p is a direction of T_{p}(S) for which the normal curvature is zero. An asymptotic curve of S is a regular connected curve C \subset S such that for each p \in C the tangent line of C at p is an asymptotic direction.

(Def) Let p be a point in S. the Dupin indicatrix at p is the set of vectors w of T_{p}(S) such that II_{p}(w) = \mp 1.

(Def 10) Let p be a point on a surface S. Two nonzero vectors w_{1}, w_{2} \in T_{p}(S) are conjugate if \langle dN_{p}(w_{1}) , w_{2} \rangle = \langle w_{1}, dN_{p} (w_{2}) \rangle = 0. Two directions r_{1}, r_{2} at p are conjugate if a pair of nonzero vectors w_{1}, w_{2} parallel to r_{1} and r_{2} , respectively, are conjugate.

(subsection 3.3.) The Gauss map in Local Coordinates

(Prop 1) Let p \in S be an elliptic point of a surface S. then there exists a neighborhood V of p in S s.t. all points in V belong to the same side of the tangent line T_{p}(S). Let p \in S be a hyperbolic point. Then in each neighborhood of p there exists points of S in both sides of T_{p}(S).

(Prop 2) Let p be a point of a surface S s.t. the Gaussian curvature K(p) \neq 0 , and let V be a connected neighborhood of p where K does not change sign. Then K(p) = \lim_{A \to 0} \frac{A’}{A} where A is the area of a region B \subset V containing p , A’ is the area of the image of B by the Gauss map N : S \to S^{2}, and the limit is taken through a sequence of regions B_{n} that converges to p, in the sense that any sphere around p contains all B_{n} for n sufficiently large.

(subsection 3.4.)Vector fields

(Def) A vector field in an open set U \subset R^{2} is a map which assigns to each q \in U a vector w(q) \in R^{2} . The vector field w is said to be differentiable if writing q = (x,y) and w(q) = (a(x,y) , b(x,y)) , the functions a and b are differentiable functions in U.

(Thm 1) Let w be a vector field in an open set U \subset R^{2}. Given p \in U, there exists a trajectory \alpha : I \to U of w (\alpha’(t) = w(\alpha(t)), t \in I) with \alpha(0) = p. This trajectory is unique in the following sense : Any other trajectory \beta : J \to U with \beta(0) = p agrees with \alpha in I \cap J.

(Thm 2) Let w be a vector field in an open set U \subset R^{2}. For each p \in U there exists a neighborhood V \subset U of p, an interval I, and a mapping \alpha : V \times I \to U s.t.

for a fixed q \in V, the curve \alpha(q,t) , t \in I, is the trajectory of w passing through q; that is, \alpha(q,0) = q, \frac{\partial \alpha}{\partial t} (q,t) = w(\alpha(q,t)).

\alpha is differentiable.

(Lem) Let w be a vector field in an open set U \subset R^{2} and let p \in U be such that w(p) \neq 0. Then there exists a neighborhood W \subset U of p and a differentiable function f: W \to R s.t. f is constant along each trajectory of w and df_{n} \new 0 for all q \in W.

(Def) A field of directions r in an open set U \subset R^{2} is a correspondence which assigns to each p \in U a line r(p) in R^{2} passing through p. r is said to be differentiable at p \in U if there exists a nonzero differentiable vector field w, defined in a neighborhood V \subset U of p, such that for each q \in V, w(q) \neq 0 is a basis of r(q); r is differentiable in U if it is differentiabl for every p \in U.

To each nonzero differentiable vector field w in U \subset R^{2}, there corresponds a differentiable field of directions given by r(p) = line generated by w(p), p \in U.

(Def 1) A vector field w in an open set U \subset S of a regular surface S is a correspondence which assigns to each p \in U a vector w(p) \in T_{p}(S). The vector field w is differentiable at p \in U if, for some parametrization x(u,v) at p, the functions a(u,v) and b(u,v) given by w(p) = a(u,v)x_{u} + b(u,v)x_{v} are differentiable functions at p; it is clear that this definition does not depend on the choice of x.

(Thm) Let w_{1} and w_{2} be two vector fields in an open set U \subset S, which are linearly independent at some point p \in U. Then it is possible to parametrize a neighborhood V \subset U of p in such a way that for each q \in V the coordinate lines of this parametrization passing through q are tangent to the lines determined by w_{1}(q) and w_{2}(q).

(Cor 1) Given two fields of directions r and r’ in an open set U \subset S s.t. at p \in U, r(p) \neq r’(p), there exists a parametrization x in a neighborhood of p s.t. the coordinate curves of x are the integral curves of r and r’.

(Cor 2) For all p \in S there exists a parametrization x(u,v) in a neighborhood V of p s.t. the coordinate curves u = const., v = const. intersect orthogonally for each q \in V (such an x is called an orthogonal parametrization).

(Cor 3) Let p \in S be a hyperbolic point of S. Then it is possible to parametrize a neighborhood of p in such a way that the coordinate curves of this parametrization are the asymptotic curves of S.

(Cor 4) Let p \in S be a nonumbilical points of S. Then it is possible to parametrize a neighborhood of p in such a way that the coordinate curves of this parametrization are the lines of curvature of S.

(subsection 3.5.) Ruled Surfaces and Minimal Surfaces

(Subsubsection A) Ruled Surfaces

(Def) A (differentiable) one-parameter family of (straight) lines [\alpha(t), w(t)] is a correspondence that assigns to each t \in I a point \alpha(t) \in R^{3} and a vector w(t) \in R^{3} , w(t) \neq 0 , so that both \alpha(t) and w(t) depend differentiably on t. For each t \in I, the line L, which passes through \alpha(t) and is parallel to w(t) is called the line of the family at t.

Given a one-parameter family of lines [\alpha(t), w(t)] , the parametrized surface x(t,v) = \alpha(t) + vw(t) , t \in I, v \in R, is called the ruled surface generated by the family [\alpha(t), w(t)]. The lines are called the rullings, and the curve \alpha(t) is called a directrix of the surface x. Sometimes we use the expression ruled surface to mean the trace of x.

(subsubsection B)Minimal Surfaces

(Def) A regular parametrized surface is called minimal if its mean curvature vanishes everywhere. A regular surface S \subset R^{3} is minimal if each of its parametrization is minimal.

Let x : U \subset R^{2} \to R^{3} be a regular parametrized surface. Choose a bounded domain D \subset U and a differentiable function h : \bar{D} \to R, where \bar{D} is the union of the domain D with its boundary \partial D. The normal variation of x(\bar{D}) , determined by h, is the map given by, \phi : \bar{D} \times (\epsilon, \epsilon) \to R^{3} : \phi (u,v,t) = x(u,v) + th(u,v)N(u,v) , (u,v) \in \bar{D} , t \in (-\epsilon, \epsilon).

The mean curvature H is H = \frac{1}{2} \frac{Eg – 2fF + Ge}{EG-F^{2}} where \langle x_{u}, N_{u} \rangle = -e , \lange x_{u} , N_{v} \rangle + \langle x_{v} , N_{u} \rangle = -2f , \langle x_{v} , N_{v} \rangle = -g.

The derivative of the area A(t) of x’(\bar{D}) at t = 0 is A’(0) = -\int_{\bar{D}}2hH \sqrt{EG-F^{2}} du dv. the area A(t) of x^{t}(\bar{D}) is A(t) = \int_{\bar{D}} \sqrt{E^{t}G^{t} – (F^{t})^{2}} du dv

(Prop 1) Let x : U \to R^{3} be a regular parametrized surface and let D \subset U be a bounded domain in U. Then x is minimal iff A’(0) = 0 for all such D and all normal variations of x(\bar{D}).

(Def) For an arbitrary parametrized regular surface, the mean curvature vector defined by \mathbf{H} = HN. A regular parametrized surface x = x(u,v) is said to be isothermal if \langle x_{u}, x_{u} \rangle = \langle x_{v} , x_{v} \rangle and \langle x_{u} , x_{v} \rangle = 0.

(Prop 2) Let x = x (u,v) be a regular parametrized surface and assume that x is isothermal. Then x_{uu} + x_{vv} = 2 \lambda^{2}\mathbf{H}, where \lambda^{2} = \langle x_{u}, x_{u} \rangle = \langle x_{v}, x_{v} \rangle.

(Def) The Laplacial \Delta f of a differentiable function f : U \subset R^{2} \to R is defined by \Delta f = (\partial^{2}f / \partial u^{2}) + (\partial^{2}f / \partial v^{2}) , (u,v) \in U. We say that f is harmonic in U if \Delta f = 0 .

(Cor) Let \mathbf{x}(u,v) = (x(u,v) , y(u,v), z(u,v)) be a parametrized surface and assume that x is isothermal. Then x is minimal iff its coordinate functions x,y,z are harmonic.

(Def) A function f : U \subset \mathbb{C} \to \mathbb{C} is analytic when, f(\zeta) = f_{1}(u,v) + i f_{2}(u,v), the real functions f_{1} and f_{2} have continuous partial derivatives of first order which satisfy the Cauchy-Riemann equations : \frac{\partial f_{1}}{\partial u} = \frac{\partial f_{2}}{\partial v}, \frac{\partial f_{1}}{\partial v} = -\frac{\partial f_{2}}{\partial u}.

Let x : U \subset R^{2} \to R^{3} be a regular parametrized surface and define complex functions \phi_{1}, \phi_{2}, \phi_{3} by \phi_{1}(\zeta) = \frac{x}{u} – i \frac{x}{v} , \phi_{2}(\zeta) = \frac{y}{u} – i \frac{y}{v} , \phi_{3}(\zeta) = \frac{z}{u} – i \frac{z}{v} , where x,y, z are the component functions of x.

(Lem) x is isothermal iff \phi_{1}^{2} + \phi_{2}^{2} + \phi_{3}^{2} \equiv 0. If this last condition is satisfied, x is minimal iff \phi_{1}, \phi_{2} and \phi_{3} are analytic functions.

(Osserman) (Thm) Let S \subset R^{3} be a regular, closed (as a subset of R^{3}) minimal surface in R^{3} which is not a plane. Then the image of the Gauss map N : S \to S^{2} is dense in the sphere S^{2} (that is, arbitrarily close to any point of S^{2} there is a point of N(S) \subset S^{2}) .

(Appendix) Self-adjoint Linear maps and Quadratic forms

(Def) V is a vector space of dimension 2, with an inner product \langle , \rangle. A linear map A : V \to V is self-adjoint if \langle Av , w \rangle = \langle v, Aw \rangle for all v , w \in V.

To each self-adjoint linear map we associate a map B : V \times V \to R defined B(v,w) = \langle Av , w \rangle. It is bilinear, and B(v,w) = B(w,v) so B is a bilinear symmetric form in V.

To each Symmetric bilinear form B in V, there corresponds a quadratic form Q in V given by Q(v) = B(v,v) , v \in V .

(Lem) If the function Q(x,y) = ax^{2} + 2bxy + cy^{2} , restricted to the unit circle x^{2} + y^{2} , has a maximum at the point (1,0), then b = 0.

(Prop) Given a quadratic form Q in V , there exists an orthonormal basis [e_{1}, e_{2}] of V s.t. if v \in V is given by v = xe_{1} + ye_{2}, then Q(v) = \lambda_{1} x^{2} + \lambda_{2} y^{2}, where \lambda_{1} and \lambda_{2} are the maximum and minimum, respectively, of Q on the unit circle |v| = 1.

(Thm) Let A : V \to V be a self-adjoint linear map. Then there exists an orthonormal basis [e_{1}, e_{2}] of V s.t. A(e_{1}) = \lambda_{1} e_{1} , A(e_{2}) = \lambda_{2} e_{2}. (that is, e_{1} and e_{2} are eigenvectors , and \lambda_{1}, \lambda_{2} are eigenvalues of A). In the basis [e_{1}, e_{2}] , the matrix of A is clearly diagonal and the elements \lambda_{1} , \lambda_{2} , \lambda_{1} \ge \lambda_{2}, on the diagonal are the maximum and the minimum, respectively, of the quadratic form Q(v) = \langle Av, v \rangle on the unit circle of V.

The Intrinsic Geometry of Surfaces

(section 4) The Intrinsic Geometry of Surfaces

(subsection 4.2) Isometries ; Conformal maps

(Def 1) S and \bar{S} will always denote regular surfaces. A diffeomorphism \phi : S \to \bar{S} is an isometry if for all p \in S and all pairs w_{1} , w_{2} \in T_{p}(S) we have \langle w_{1}, w_{2} \rangle_{p} = \langle d\phi_{p} (w_{1}) , d\phi_{p}( w_{2}) \rangle_{\phi(p)} , the surfaces S and \bar{S} are then said to be isometric.

(Def 2) A map \phi : V \to \bar{S} of a neighborhood V of p \in S is a local isometry at p if there exists a neighborhood \bar{C} of \phi(p) \in \bar{S} s.t. \phi : V \to \bar{V} is an isometry. If there exists a local isometry into \bar{S} at every p \in S, the surface S is said to be locally isometric to \bar{S} . S and \bar{S} are locally isometric if S is locally isometric to \bar{S} and \bar{S} is locally isometric to S.

(Prop 1) Assume the existence of parametrizations x : U \to S and \bar{x} : U \to \bar{S} s.t. E = \bar{E}, F = \bar{F}, G = \bar{G} in U. Then the map \phi = \bar{x} \bullet x^{-1} ; x(U) \to \bar{S} is a local isometry.

(Def 3) A diffeomorphism \phi : S \to \bar{S} is called a conformal map if for all p \in S and all v_{1} , v_{2} \in T_{p}(S) we have \langel d\phi_{p}(v_{1}, d\phi_{p}(v_{2}) \rangle = \lambda^{2}(p) \langle v_{1}, v_{2} \rangle_{p} , where \lambda^{2} is a nowhere-zero differentiable function on S; the surfaces S and \bar{S} are then said to be conformal. A map \phi : V \to \bar{S} of a neighbhorhood V of p \in S into \bar{S} is a local conformal map at p if there exists a neighborhood \bar{V} of \phi(p) s.t. \phi : V \to \bar{V} is a conformal map. If for each p \in S, there exists a local conformal map at p, the surface S is said to be locally conformal to \bar{S}.

(Prop 2) Let x : U \to S and \bar{x} : U \to \bar{S} be parametrizations s.t. E = \lambda^{2} \bar{E}, F = \lambda^{2} \bar{F}, G = \lambda^{2} \bar{G} in U, where \lambda^{2} is a nowhere-zero differentiable function in U. Then the map \phi = \bar{x} \bullet x^{-1} : x (U) \to \bar{S} is a local conformal map.

(Thm) Any two regular surfaces are locally conformal.

(subsection 4.3.) The Gauss Theorem and the Equations of Compatibility

(Def) Given x : U \subset R^{2} \to S be a parametrization in the orientation of S. By expressing the derivatives of the vectors x_{u}, x_{v} and N in the basis [x_{u}, x_{v} , N] , we obtain

x_{uu} = \Gamma_{1}^{1}{1} x{u} + \Gamma_{1}^{2}{1} x{v} + L_{1}N ,

x_{uv} = \Gamma_{1}^{1}{2} x{u} + \Gamma_{1}^{2}{2} x{v} + L_{2}N ,

x_{vu} = \Gamma_{2}^{1}{1} x{u} + \Gamma_{2}^{2}{1} x{v} + L_{3}N ,

x_{vv} = \Gamma_{2}^{1}{2} x{u} + \Gamma_{2}^{2}{2} x{v} + L_{4}N ,

N_{u} = a_{1 1} x_{u} + a_{21} x_{v} ,

N_{v} = a_{12} x_{u} + a_{22} x_{v}.

where the a_{ij} , i,j = 1,2, were obtained in Chap. 3 and the other coefficients are to be determined. The coefficients \Gamma_{ij}^{k} ,i,j,k = 1,2, are called the Christoffel symbols of S in the parametrization x. Since x_{uv} = x_{vu} , \Gamma_{12}^{1} = \Gamma_{21}^{1} and \Gamma_{12}^{2} = \Gamma_{21}^{2}.

We obtain L_{1} = e , L_{2} = \bar{L_{2}} = f, L_{3} = g, where e,f,g are coefficients of the second fundamental form of S.

All geometric concepts and properties expressed in terms of Christoffel symbols are invariant under isometries.

(Theorema Egregium) (Gauss) The Gaussian curvature K of a surface is invariant by local isometries.

(Def) Gauss formula : (\Gamma_{12}^{2}){u} – (\Gamma{11}^{2}){v} + \Gamma{12}^{1}\Gamma_{11}^{2} +\Gamma_{12}^{2}\Gamma_{12}^{2} - \Gamma_{11}^{2}\Gamma_{22}^{2} - \Gamma_{11}^{1}\Gamma_{12}^{2} = -E \frac{eg-f^{2}}{EG-F^{2}} = -EK.

(Def) Mainardi-Codazzi equations

e_{v} – f_{u} = e\Gamma_{12}^{1} + f(\Gamma_{12}^{2} - \Gamma_{11}^{1}) – g \Gamma_{11}^{2}

f_{v} – g_{u} = e \Gamma_{22}^{1} + f(\Gamma_{22}^{2} - \Gamma_{12}^{1} ) -g \Gamma_{12}^{2}

(Bonnet) (Thm) Let E,F,G,e,f,g, be differentiable functions, defined in an open set V \subset R^{2} , with E >0, and G >0. Assume that the given functions satisfy formally the Gauss and Mainardi-Codazzi equations and that EG-F^{2} >0. Then, for every q \in V there exists a neighborhood U \subset V of q and a diffeomorphism x : U \to x(U) \subset R^{3} s.t. the regular surface x(U) \subset R^{3} has E,F,G, and e,f,g, as coefficients of the first and second fundamental forms, respectively. Furthermore, if U is connected and if \bar{x} : U \to \bar{x}(U) \subset R^{3} is another diffeomorphism satisfying the same conditions, then there exist a translation T and a proper linear orthogonal transformation \rho in R^{3} s.t. \bar{x} = T \bullet \rho \bullet x.

(subsection 4.4.) Parallel Transport . Geodesics.

(Def 1) Let w be a differentiable vector field in an open set U \subset S and p \in U. Let y \in T_{p}(S). Consider a parametrized curve \alpha : (-\epsilon , \epsilon) \to U, with \alpha(0) = p and \alpha’(0) = y, and let w(t) , t \in (- \epsilon, \epsilon) , be the restriction of the vector field w to the curve \alpha. The vector obtained by the normal projection of (dw/dt)(0) onto the plane T_{p}(S) is called the covariant derivative at p of the vector field w relative to the vector y. This covariant derivative is denoted by (Dw/dt)(0) or (D,w)(p)

(Def 2) A parametrized curve \alpha : [0,l] \to S is the restriction to [0,l] of a differentiable mapping of (0 - \epsilon , l + \epsilon) , \epsilon >0, into S. If \alpha(0) = p and \alpha(l) = q, we say that \alpha joins p to q. \alpha is regular if \alpha’(t) \neq 0 for t \in [0, l].

(Def 3) Let \alpha : I \to S be a parametrized curve in S. A vector field w along \alpha is a correspondence that assigns to each t \in I a vector w(t) \in T_{\alpha(t)} (S). The vector field w is differentiable at t_{0} \in I if for some parametrization x(u,v) in \alpha(t_{0}) the components a(t), b(t) of w(t) = ax_{u} + bx_{v} are differentiable functions of t at t_{0}. w is differentiable in I if it is differentiable for every t \in I.

(Def 4) Let w be a differentiable vector field along \alpha : I \to S. The expression

\frac{Dw}{dt} = (a’ + \Gamma_{11}^{1} au’ + \Gamma_{12}^{1} av’ + \Gamma_{12}^{1}bu’ + \Gamma_{22}^{1}bv’) x_{u} + (b’ + \Gamma_{11}^{2}au’ + \Gamma_{12}^{2} av’ + \Gamma_{12}^{2}bu’ + \Gamma_{22}^{2} bv’) x_{v}

of (Dw/dt)(t), t \in I , is well-defined and is called the covariant derivative of w at t.

(Def 5) A vector field w along a parametrized curve \alpha : i \to S is said to be parallel if Dw/dt = 0 for every t \in I.

(prop 1) Let w and v be parallel vector fields along \alpha : I \to S. Then \langle w(t), v(t) \rangle is constant. In particular, |w(t)| and |v(t)| are constant, and the angle between v(t) and w(t) is constant.

(prop 2) Let \alpha : I \to S be a parametrized curve in S and let w_{0} \in T_{\alpha(t_{0}) (S) , t_{0} \in I. Then there exists a unique parallel vector field w(t) along \alpha(t) , with w(t_{0}) = w_{0}.

(Def 6) Let \alpha : I \to S be a parametrized curve and w_{0} \in T_{\alpha(t_{0})} (S) , t_{0} \in I. Let w be the parallel vector field along \alpha, with w(t_{0}) = w_{0}. The vector w(t_{1}) , t_{1} \in I, is called the parallel transport of w_{0} along \alpha at the point t_{1}.

(Def 7) A map \alpha : [0,l] \to S is a parametrized piecewise regular curve if \alpha is continuous and there exists a subdivision 0 = t_{0} < t_{1} < \cdots < t_{k} < t_{k+1} = l of the interval [0,l] in such a way that the restriction \alpha | [t_{i}, t_{i+1}] , i = 0, …, k is a parametrized regular curve. Each \alpha|[t_{i},t_{i+1}] is called a regular arc of \alpha .

(Def 8) A nonconstant, parametrized curve \gamma : I \to S is said to be geodesic at t \in I if the field of tangent vectors \gamma ‘(t) is parallel along \gamma at t; that is , \frac{D\gamma’(t)}{dt} = 0 ; \gamma is a parametrized geodesic if it is geodesic for all t \in I.

(Def 8a) A regular connected curve C in S is said to be a geodesic if , for every p \in S, the parametrization \alpha(s) of a coordinate neighborhood of p by the arc length s is a parametrized geodesic; that is, \alpha’(s) is a parallel vector field along \alpha(s).

(Def 9) Let w be a differentiable field of unit vectors along a parametrized curve \alpha : I \to S on an oriented surface S. Since w(t), t \in I, is a unit vector field, (dw/dt)(t) is a normal to w(t), and therefore \frac{Dw}{dt} = \lambda(N \wedge w(t)) . The real number \lambda = \lambda(T) , denoted by [Dw/dt] , is called the algebraic value of the covariant derivative of w at t.

(Def 10) Let C be an oriented regular curve contained on an oriented surface S ,and let \alpha(s) be a parametrization of C, in a neighborhood of p \in S, by the arc length s. The algebraic value of the covariant derivative [D\alpha’(s)/ds] = k_{g} of \alpha’(s) at p is called the geodesic curvature of C at p.

(Lem 1) Let a and b be differentiable functions in I with a^{2} + b^{2} = 1 and \phi_{0} be s.t. a(t_{0}) = cos \phi_{0} , b(t_{0}) = sin \phi_{0} . Then the differentiable function \phi = \phi_{0} + \int_{t_{0}}^{t} (ab’ – ba’) dt is such that cos \phi(t) = a(t), sin \phi(t) = b(t) , t \in I, and \phi(t_{0}) = \phi_{0}.

(Lem 2) Let v and w be two differentiable vector fields along the curve \alpha : I \to S, with |w(t)| = |v(t)| = 1, t \in I, Then [\frac{Dw}{dt}] – [\frac{Dv}{dt}] = \frac{d\phi}{dt} where \phi is one of the differentiable determinations of the angle fron v to w, as given by (Lem 1).

(Prop 3) Let x(u,v) be an orthogonal parametrization (that is, F = 0) of a neighborhood of an oriented surface S, and w(t) be a differentiable field of unit vectors along the curve x(u(t), v(t)). Then [\frac{Dw}{dt}] = \frac{1}{2\sqrt{EG}} {G_{u}\frac{dv}{dt} – E_{v}\frac{du}{dt}} + \frac{d\phi}{dt} where \phi(t) is the angle from x_{u} to w(t) in the given orientation.

(Liouville)(prop 4) Let \alpha(s) be a parametrization by arc length of a neighborhood of a point p \in S of a regular oriented curve C on an oriented surface S. Let x(u,v) be an orthogonal parametrization of S in p and \phi(s) be the angle that x_{u} makes with \alpha’(s) in the given orientation. Then k_{g} = (k_{g}){1} cos \phi + (k{g}){2} sin \phi + \frac{d\phi}{ds} where (k{g}){1} and (k{g})_{2} are the geodesic curvatures of the coordinate curves v = const. and u = const. respectively.

(Prop 5) Given a point p \in S and a vector w \in T_{p}(S), w \neq 0 . There exists an \epsilon >0 and a unique parametrized geodesic \gamma : (-epsilon , \epsilon) \to S s.t. \gamma(0) = p , \gamma’(0) = w.

(subsection 4.5.) The Gauss-Bonnet Theorem and its Applications

(Def) Let \alpha : [0,l] \to S be a continuous map from the closed interval [0,l] into the regular surface S. We say that \alpha is a simple, closed, piecewise regular, parametrized curve if

\alpha(0) = \alpha(l).

t_{1} \neq t_{2}, t_{1}, t_{2} \in [0,l) implies that \alpha(t_{1}) \neq \alpha(t_{2}) .

There exists a subdivision 0 = t_{0} < t_{1} < \cdots < t_{k} < t_{k+1} = l of [0,l] s.t. \alpha is differentiable and regular in each [t_{i}, t_{i+1}] i = 0,…,k.

(Def) \alpha is a closed curve which fails to have a well-defined tangent line only at a finite number of points. The points \alpha(t_{i}) , i = 0,…,k are called the vertices of \alpha and the traces \alpha([t_{i}, t_{i+1}]) are called the regular arcs of \alpha. Assume now that S is oriented and let |\theta_{i}| , 0 < |\theta_{i}| \le \pi , be the smallest determination of the angle from \alpha’(t_{i} – 0) to \alpha’(t_{i} + 0) . we give \theta_{i} the sign of the determinant (\alpha’(t_{i} – 0) , \alpha’(t_{i} + 0), N).

(Def) Let x : U \subset R^{2} \to S be a parametrization compatible with the orientation of S. Assume that U is homeomorphic to an open disk in the plane.

Let \alpha : [0,l] \to x(U) \subset S be a simple closed, piecewise regular, parametrized curve, with vertices \alpha(t_{i}) and external angles \theta_{i}, i = 0,…,k.

Let \phi_{i} : [t_{i}, t_{i+1}] \to R be differentiable functions which measure at each t \in [t_{i}, t_{i+1}] the positive angle from x_{u} to \alpha’(t) (section 4-4 Lem 1)

(Thm)(Of Turning Tangents) With the above notation \sum_{i = 0}^{k} (\phi_{i}(t_{i+1}) - \phi_{i}(t_{i})) + \sum_{i = 0}^{k} \theta_{i} = \mp 2\pi where the sign plus or minus depends on the orientation of \alpha.

(Def) Let x : U \subset R^{2} \to S be a parametrization of S compatible with its orientation and let R \subset x(U) be a bounded region of S. If f is a differentiable function on S, the integral \int \int_{x^{-1}(R)} f(u,v) \sqrt{EG – F^{2}} du dv does not depend on the parametrization x, chosen in the class of orientation of x. It is called the integral of f over the region R. Denote it by \int \int_{R} f d \sigma.

(Local)(Gauss Bonnet Theorem) Let x : U \to S be an orthogonal parametrization (that is , F = 0) , of an oriented surface S, where U \subset R^{2} is homeomorphic to an open disk and x is compatible with the orientation of S. Let R \subset x(U) be a simple region of S and let \alpha : I \to S be s.t. \partial R = \alpha(I). Assume that \alpha is positively oriented, parametrized by arc length s, and let \alpha(s_{0}), \alpha(s_{k}) and \theta_{0} , …, \theta_{k} be, respectively, the vertices and the external angles of \alpha. Then \sum_{i = 0}^{k} \int_{s_{i}}^{s_{i+1}} k_{g}(s) ds + \int \int_{R}K d \sigma + \sum_{i = 0}^{k} \theta_{o} = 2 \pi where k_{g}(s) is the geodesic curvature of the regular arcs of \alpha and K is the Gaussian curvature of S.

(Def) Let S be a regular surface. A region R \subset S is said to be regular if R is compact and its boundary \partial R is the finite union of (simple) closed piecewise regular curves which do not intersect . We shall consider a compact surface as a regular region, the boundary of which is empty.

A simple region which has only three vertices with external angles \alpha_{i} \neq 0 , i = 1,2,3, is called a triangle.

A triangulation of a regular region R \subset S is a finite family \mathfrak{J} of triangles T_{i}, i = 1,….,n, s.t.

\bigcup_{i=1}^{n} T_{i} = R.

If T_{i} \cap T_{j} \neq \empty, then T_{i} \cap T_{j} is either a common edge of T_{i} and T_{j} or a common vertex of T_{i} and T_{j}.

Given a triangulation \mathfrak{J} of a regular region R \subset S of a surface S, denote F the number of triangles(faces), by E the number of sides(edges), by V the number of vertices of the triangulation. The number F-E+V = \chi is called The Euler-Poincare characteristic of the triangulation.

(prop 1) Every regular region of a regular surface admits a triangulation.

(prop 2) Let S be an oriented surface and {x_{\alpha}}, \alpha \in A, a family of parametrizations compatible with the orientation of S. Let R \subset S be a regular region of S. Then there is a triangulation \mathfrak{J} of R s.t. every triangle T \in \mathfrak{J} is contained in some coordinate neighborhood of the family {x_{\alpha}} . Furthermore, If the boundary of every triangle of \mathfrak{J} is positively oriented , adjacent triangles determine opposite orientations in the common edge.

(Prop 3) If R \subset S is a regular region of a surface S, the Euler-Poincare characteristic does not depend on the triangulation of R. It is convenient, therefore, to denote it \chi(R).

(Prop 4) Let S \subset R^{3} be a compact connected surface; then one of the values 2,0,-2,…,-2n, …, is assumed by the Euler-Poincare characteristic \chi(S). Furthermore, if S’ \subset R^{3} is another compact surface and \chi(S) = \chi(S’), then S is homeomorphic to S’.

(Prop 5) Let R \subset S be a regular region of an oriented surface S and let \mathfrak{J} be a triangulation of R s.t. every triangle T_{j} \in \mathfrak{J} , j = 1,…,k, is contained in a coordinate neighborhood x_{j}(U_{j}) of a family of parametrizations {x_{\alpha}} , \alpha \in A, compatible with the orientation of S. Let f be a differentiable function on S. the sum \sum_{j = 1}^{k} \int \int_{x_{j}^{-1}(T_{j}) f(u_{j}, v_{j}) \sqrt{E_{j}G_{j} – F_{j}^{2}} du_{j} dv_{j} does not depend on the triangulation \mathfrak{J} or on the family {x_{j}} of parametrizations of S.

(Global Gauss-Bonnet Thm) Let R \subset S be a regular region of an oriented surface and let C_{1}, …, C_{n} be the closed, simple, piecewise regular curves which form the boundary \partial R of R. Suppose that each C_{i} is positively oriented and let \theta_{1}, …, \theta_{p} be the set of all external angles of the curves C_{1}, …, C_{n} . Then \sum_{i = 1}^{n} \int_{C_{i}} k_{g}(s) ds + \int \int_{R} K d \sigma + \sum_{i = 1}^{p} \theta_{i} = 2 \pi \chi(R), where s denotes the arc length of C_{i}, and the integral over C_{i} means the sum of integrals in every regular arc of C_{i}.

(Cor 1) If R is a simple region of S, then \sum_{i = 0}^{k} \int_{s_{i}}^{s_{i+1}} k_{g}(s) ds + \int \int_{R} k d \sigma + \sum_{i = 0}^{k} \theta_{i} = 2\pi.

(Cor 2) Let S be an orientable compact surface ; then \int \int_{S} K d \sigma = 2 \pi \chi(S).

(Prop) Assume Every piecewise regular curve in the plane(thus without self-intersections) is the boundary of a simple region.

A compact surface of positive curvature is homeomorphic to a sphere.

Let S be an orientable surface of negative or zero curvature. Then two geodesics \gamma_{1} and \gamma_{2} which start from a point p \in S cannot mmet again at a point q \in S in such a way that the traces of \gamma_{1} and \gamma_{2} constitute the boundary of a simple region R of S.

Let S be a surface homeomorphic to a cylinder with Gaussian curvature K < 0. Then S has at most one simple closed geodesic.

If there exist two simple closed geodesics \Gamma_{1} and \Gamma_{2} on a compact surface S of a positive curvature, then \Gamma_{1} and \Gamma_{2} intetsect.

Let \alpha : I \to R^{3} be a closed, regular, parametrized curve with nonzero curvature. Assume that the curve described by normal vector n(s) in the unit sphere S^{2} (the normal indicatrix) is simple. Then n(I) divides S^{2} in two regions with equal areas.

Let T be a geodesic triangle(that is, the sides of T are geodesics) in an oriented surface S. Let \theta_{1}, \theta_{2}, \theta_{3} be the external angles of T and let \phi_{1} = \pi - \theta_{1}, \phi_{2} = \pi -\theta_{2}, \phi_{3} = \pi - \theta_{3} be its interior angles. \int \int_{T} K d \sigma = - \pi + \sum_{i = 1}^{3} \phi_{i}. The sum of the interior angles , \sum_{i = 1}^{3} \phi_{i} of a geodesic triangle is Equal to \phi if K = 0, Greater than \pi if K > 0 , Smaller that \pi if K < 0.

Furthermore, the difference - \pi + \sum_{i = 1}^{3} \phi_{i} (the excess of T) is given precisely by \int \int_{T} K d \sigma. If K \neq 0 on T, this is the area of the image N(T) of T by the Gauss map N : S \to S^{2}. In other words, The excess of a geodesic triangle T is equal to the area of its spherical image N(T).

Vector fields on surfaces. Let v be a differentiable vector field on an oriented surface S. We say that p \in S is a singular point of v if v(p) = 0. The singular point p is isolated if there exists a neighborhood V of p in S such that v has no singular points in V other than p. let x : U \to S be an orthogonal parametrization at p = x(0,0) compatible with the orientation of S, and let \alpha : [0,l] \to S be a simple, closed, piecewise regular parametrized curve s.t. \alpha([0,l]) \subset x(U) is the boundary of a simple region R containing p as its only singular point. Let v = v(t), t \in [0,l] , be the restriction of v along \alpha, and let \phi = \phi(t) be some differentiable determination of the angle from x_{u} to v(t), given by (Lem 1 of Sec 4.4) Since \alpha is closed, there is an integer I defined by 2\pi I = \phi(l) - \phi(0) = \int_{0}^{t} \frac{d\phi}{dt} dt . I is called the index of v at p. (Poincare’s Theorem) The sum of the indices of a differentiable vector field v with isolated singular points on a compact surface S is equal to the Euler-Poincare characteristic of S.

(subsection 4.6.) The exponential map. Geodesic Polar Coordinates

(Lem 1) If the geodesic \gamma(t,v) is defined for t \in (-\epsilon , \epsilon) , then the geodesic \gamma (t, \lambda v) , \lambda \in R, \lambda \neq 0, is defined for t \in (-\epsilon / \lambda , \epsilon / \lambda) , and \gamma(t, \lambda v) = \gamma(\lambda t , v) .

(Def) If v = T_{p}(S), v \neq 0, is s.t. \gamma(|v| , v/|v|) = \gamma(1,v) is defined, we set exp_{p}(v) = \gamma(1,v) and exp_{p}(0) = p.

(prop 1) Given p \in S there exists an \epsilon >0 s.t. exp_{p} is defined and differentiable in the interior B_{e} of a disk of radius \epsilon of T_{p}(S), with center in the origin.

(prop 2) exp_{p} : B_{e} \subset T_{p}(S) \to S is a diffeomorphism in a neighborhood U \subset B , of the origin 0 of T_{p}(S).

(Def) V \subset S a normal neighborhood of p \in S if V is the image V = exp_{p}(U) of a neighborhood U of the origin of T_{p}(S) restricted to which exp_{p} is a diffeomorphism.

Since the exponential map at p \in S is a diffeomorphism on U, it may be used to introduce coordinates in V. The normal coordinates which correspond to a system of rectangular coordinates in the tangent plane T_{p}(S) . The geodesic polar coordinates which correspond to polar coordinates in the tangent plane T_{p}(S).

polar coordinates in the plane are not defined in the closed half-line l which corresponds to \theta = 0. Set exp_{p}(l) = L . Since exp_{p} : U-l \to V-L is still a diffeomorphism, we may parametrize the points of V -L by the coordinates (\rho , \theta), which are called geodesic polar coordinates.

The images by exp_{p} : U \to V of circles in U centered in 0 will be called geodesic circles of V, and the imagnes of exp_{p} of the lines through 0 will be called radial geodesics of V.

(prop 3) Let x : U -l \to V-L be a system of geodesic polar coordinates (\rho , \theta). Then the coefficients E = E(\rho, \theta) , F = F(\rho, \theta) , and G = G(\rho, \theta) of the first fundamental form satisfy the conditions E = 1, F = 0, \lim_{\rho \to 0} G = 0, \lim_{\rho \to 0} (\sqrt(G))_{\rho} = 1.

(Minding)(Thm) Any two regular surfaces with the same constant Gaussian curvature are locally isometric. More precisely, Let S_{1}, S_{2} be two regular surfaces with the same constant curvature K. Choose points p_{1} \in S_{1} , p_{2} \in S_{2} , and orthonormal basis {e_{1}, e_{2}} \in T_{p_{1}} (S_{1}) , [f_{1}, f_{2}] \in T_{p_{2}} (S_{2}) . Then there exists neighborhoods V_{1} of p_{1} , V_{2} of p_{2} and an isometry \psi : V_{1} \to V_{2} s.t. d\psi(e_{1}) = f_{1}, d\psi(e_{2}) = f_{2}.

(prop 4) Let p be a point on a surface S. Then, there exists a neighborhood W \subset S of p s.t. if \gamma : I \to W is a parametrized geodesic with \gamma(0)= p, \gamma(t_{1}) = q, t_{1} \in I, and \alpha : [0,t_{1}] \to S is a parametrized regular curve joining p to q, we have l_{\gamma} \le l_{\alpha} , where I_{\alpha} denotes the length of the curve \alpha. Moreofer, if I_{\gamma} = I_{\alpha} , then the trace of \alpha coincides with the trace of \alpha between p and q.

(Prop 5) Let \alpha : I \to S be a regular parametrized curve with a parameter proportional to arc length. Suppose that the arc length of \alpha between any two points t, \tau \in I, is smaller than or equal to the arc length of any regular parametrized curve joining \alpha(t) to \alpha(\tau) . Then \alpha is a geodesic.

(Thm 1) Let S a regular surface. Given p \in S there exist numbers \epsilon_{1} >0 , \epsilon_{2} >0 and a differentiable map \gamma : (-\epsilon_{2}, \epsilon_{2}) \times B_{\epsilon_{1}} \to S, B_{\epsilon_{1}} \subset T_{p}(S) s.t. for v \in B_{\epsilon_{1}} , v \neq 0 , t \in (-\epsilon_{2}, \epsilon_{2}) the curve t \in \gamma(t,v) is the geodesic of S with \gamma(0,v) = p , \gamma’(0,v) = v and for v = 0 \gamma(t,0) = p.

(Thm 1a.) Given p \in S, there exists positive numbers \epsilon, \epsilon_{1}, \epsilon_{2} and a differentiable map \gamma : (-\epsilon_{2}, \epsilon_{2}) \times \mathcal{U} \to S where \mathcal{U} = {(q,v) ; q \in B_{\epsilon}(p), v \in B_{\epsilon_{1}}(0) \subset T_{q}(S)}, s.t. \gamm(t,q,0) =q, and for v \neq 0 the curve t \to \gamma(t,q,v), t \in (-\epsilon_{2}, \epsilon_{2}) is the geodesic of S with \gamma(0,q,v) = q, \gamma’(0,q,v) = v.

(prop 1) Given p \in S there exist a neighborhood W of p in S and a number \delta >0 s.t. for every q \in W, exp_{q} is a diffeomorphism on B_{\delta}(0) \subset T_{q}(S) and exp_{q}(B_{\delta}(0)) \supset W ; that is, W is a normal neighborhood of all its points.

(prop 2) Let \alpha : I \to S be a parametrized, piecewise regular curve s.t. in each regular arc the parameter is proportional to the arc length. Suppose that the arc length between any two of this points is smaller than or equal to the arc length of any parametrized regular curve joining these points. Then \alpha is a geodesic ; in particular, \alpha is regular everywhere.

(prop 3) For each point p \in S there exists a positive number \epsilon with the following property ; If a geodesic \gamma(t) is tangent to the geodesic circle S_{r}(p), r < \epsilon , at \gamma(0) , then for t \neq 0 small, \gamma(t) is outside B_{t}(p) .

(Existence of Convex neighborhoods) (prop 4) For each point p \in S there exists a number c > 0 s.t. B_{c}(p) is convex ; that is, any two points of B_{c}(p) can be joined by a unique minimal geodesic in B_{c}(p) .

Global Differential Geometry

(Section 5) Global Differential Geometry

(subsection 5.2.) The Rigidity of the Sphere

(Thm 1) Let S be a compact , connected, regular surface with constant Gaussian curvature K. Then S is a sphere.

(Lem 1) Let S be a regular surface and p \in S a point of S satisfying the following conditions :

K(p) >0 ; that is, the Gaussian curvature in p is positive.

p is simultaneously a point of local maximum for the function k_{1} an a point of local minimum for the function k_{2} (k_{1} \ge k_{2}) .

Then p is an umbilical point of S.

(Thm 1a.) Let S be a regular, compact, and connected surface with Gaussian curvature K > 0 and mean curvature H constant. Then S is a sphere.

(Thm 1b) Let S be a regular, compact, and connected surface of positive Gaussian curvature. If there exists a relation k_{2} = f(k_{1}) in S, where f is a decreasing function of k_{1}, k_{1} \ge k_{2}, then S is a sphere.

(Lem 2) A regular compact surface S \subset R^{3} has at least one elliptic point.

(subsection 5.3.) Complete Surfaces . Theorem of Hopf-Rinow

(Def 1) A regular (connected) surface S is said to be extendable if there exists a regular (connected) surface \bar{S} s.t. S \subset \bar{S} as a proper subset. If there exists no such \bar{S}, S said to be nonextendable.

(Def 2) A regular surface S is said to be complete when for every point p \in S, any parametrized geodesic \gamma : [0, \epsilon) \to S of S, starting from p = \gamma(0) , may be extended into a parametrized geodesic \bar{\gamma} : R \to S, defined on the entire line R. In other words, S is complete when for every p \in S the mapping exp_{p} : T_{p} (S) \to S is defined for every v \in T_{p}(S).

(Prop 1) A complete surface S is nonextendable.

(Prop 2) Given two points p, q \in S of a regular(connected) surface S, there exists a parametrized piecewise differentiable curve joining p to q.

(Def 3) The (intrinsic) distance d(p,q) from the point p \in S to the point q \in S is the number d(p,q) = \inf 1(\alpha_{p,q}) , where the inf is taken over all piecewise differentiable curves joining p to q.

(Prop 3) The distance d defined above has the following properties.

d(p,q) = d(q,p),

d(p,q) + d(q,r) \ge d(p,r) ,

d(p,q) \ge 0,

d(p,q) = 0 iff p = q.

(Cor) |d(p,r) – d(r,q)| \le d(p,q).

(prop 4) If we let p_{0} \in S be a point of S, then the function f : S \to R given by f(p) = d(p_{0}, p) , p \in S, is continuous on S.

(prop 5) A closed surface S \subset R^{3} is complete.

(Cor) A compact surface is complete.

(Def) A geodesic \gamma joining two points p, q \in S is minimal if its length l(\gamma) is smaller than or equal to the length of any piecewise regular curve joining p to q.

(Hopf-Rinow) (Thm) Let S be a complete surface. Given two points p, q \in S, there exists a minimal geodesic joining p to q.

(Cor 1) Let S be complete. Then for every point p \in S the map exp_{p} : T_{p}(S) \to S is onto S.

(Cor 2) Let S be complete and bounded in the metric d (that is, there exists r >0 s.t. d(p,q) < r for every pair p,q \in S). Then S is compact.

(subsection 5.4.) First and Second Variations of Arc length ; Bonnet’s Theorem

(Def 1) Let \alpha : [0,l] \to S be a regular parametrized curve, where the parameter s \in [0,l] is the arc length. A variation of \alpha is a differentiable map h : [0,l] \times (-\epsilon, \epsilon) \subset R^{2} \to S s.t. h(s,0) = \alpha(s), s \in [0,l] . For each t \in (-\epsilon, \epsilon) , the curve h_{t} : [0,l] \to S, given by h_{t}(s) = h(s,t), is called a curve of the variation h. A variation h is said to be proper if h(0,t) = \alpha(0) , h(l,t) = \alpha(l), t \in (-\epsilon, \epsilon).

(Def) A variation h of \alpha determines a differentiable vector field V(s) along \alpha by V(s) = \frac{\partial h}{\partial s}(s,0) , s \in [0,l] . V is called the variational vector field of h; If h is proper, then V(0) = V(l) = 0.

(prop 1) If we let V(s) be a differentiable vector field along a parametrized regular curve \alpha : [0,l] \to S then there exists a variation h : [0,l] \times (-\epsilon , \epsilon) \to S of \alpha s.t. V(s) is the variational vector field of h. Furthermore, if V(0) = V(l) = 0, then h can be chosen to be proper.

(Def) Define a function L : (-\epsilon, \epsilon) \to R by L(t) = \int_{0}^{l} |\frac{\partial h}{\partial s}(s,t)|ds, t \in (-\epsilon, \epsilon) .

(Lem 1) The function L defined above is differentiable in a neighborhood of t= 0 ; in such a neighborhood , the derivative of L may be obtained by differentiation under the integral sign.

(Lem 2) Let w(t) be a differentiable vector field along the parametrized curve \alpha : [a,b] \to S and let f : [a,b] \to R be a differentiable function. Then \frac{D}{dt}(f(t)w(t)) = f(t) \frac{Dw}{dt} + \frac{df}{dt}w(t).

(Lem 3) Let v(t) and w(t) be differentiable vector fields along the parametrized curve \alpha : [a,b] \to S. Then \frac{d}{dt} \langle v(t), w(t) \rangle = \langle \frac{Dv}{dt},w(t) \rangle + \langle v(t), \frac{Dw}{dt} \rangle.

(Lem 4) Let h : [0,l] \times (-\epsilon, \epsilon) \subset R^{2} \to S be a differentiable mapping. Then \frac{D}{\partial s} \frac{\partial h}{\partial t} (s,t) = \frac{D}{\partial t} \frac{\partial h}{\partial s} (s,t).

(prop 2) Let h : [0,l] \times (-\epsilon, \epsilon) be a proper variation of the curve \alpha : [0,l] \to S and let V(s) = (\partial h / \partial t)(s,0) , s \in [0,l] , be the variational vector field of h. Then L’(0) = -\int_{0}^{l} \langle A(s) , V(s) \rangle ds, where A(s) = (D/\partial s) (\partial h / \partial s) (s,0).

(prop 3) A regular parametrized curve \alpha : [0,l] \to S , where the parameter s \in [0,l] is the arc length of \alpha , is a geodesic iff , for every proper variation h : [0,l] \times (-\epsilon , \epsilon) \to S of \alpha , L’(0) = 0.

(Lemma 5) Let x : U \to S be a parametrization at a point p \in S of a regular surface S, with parameters u,v, and let K be a Gaussian curvature of S. Then \frac{D}{\partial v}\frac{D}{\partial u} x_{u} - \frac{D}{\partial u}\frac{D}{\partial v}x_{u} = K(x_{u} \wedge x_{v}) \wedge x_{u}.

(Lem 6) Let h : [0,l] \times (-\epsilon, \epsilon) \to S be a differentiable mapping and let V(s,t) , (s,t) \in [0,l] \times (-\epsilon , \epsilon) , be a differentiable vector field along h. Then \frac{D}{\partial t}\frac{D}{\partial s} V - \frac{D}{\partial s}\frac{D}{\partial t}V = K(s,t)(\frac{\partial h}{\partial s} \wedge \frac{\partial h}{\partial t}) \wedge V, where K(s,t) is the curvature of S at the point h(s,t).

(Prop 4) Let h : [0,l] \times (-\epsilon , \epsilon) \to S be a proper orthogonal variation of a geodesic \gamma :[0,l] \to S parametrized by the arc length s \in [0,l] . Let V(s) = (\partial h/\partial t) (s,0) be the variational vector field of h. Then L’’(0) = \int_{0}^{t} (|\frac{D}{\partial s}V(s)|^{2} -K(s)|V(s)|^{2})ds where K(s) = K(s,0) is the Gaussian curvature of S at \gamma(s) = h(s,0).

(Bonnet)(Thm) Let the Gaussian curvature K of a complete surface S satisfy the condition K \ge \delta > 0 . Then S is compact and the diameter \rho of S satisfies the inequality \rho \le \frac{\pi}{\sqrt{\delta}} .

(subsection 5.5.) Jacobi Fields and Conjugate Points

(Def 1) Let \gamma : [0,l] \to S be a parametrized geodesic on S and let h : [0,l] \times (-\epsilon , \epsilon) \to S be a variation of \gamma s.t. for every t \in (-\epsilon , \epsilon) the curve h_{t}(s) = h(s,t) , s \in [0,l] , is a parametrized geodesic (not necessarily parametrized by arc length) . The variational field (\partial h/\partial h)(s,0) = J(s) is called a Jacobi field along \gamma.

(Prop 1) Let J(s) be a Jacobi field along \gamma : [0,l] \to S, s \in [0,l] . Then J satisfies the so-called Jacobi equation \frac{D}{ds}\frac{D}{ds}J(s) + K(s)(\gamma’(s) \wedge J(s)) \wedge \gamma’(s) = 0, where K(s) is the Gaussian curvature of S at \gamma(s).

(Lem 1) Let p \in S and choose v,w \in T_{p}(S), with |v| = l. Let \gamma:[0,l] \to S be the geodesic on S given by \gamma(s) = exp_{p}(sv) , s \in [0,l]. Then, the vector field J(s) along \gamma given by J(s) = s(d exp_{p})_{sv}(w) , s \in [0,l] , is a Jacobi field. Furthermore, J(0) = 0, (DJ/ds) (0) = w.

(Prop 2) If we let J(s) be a differentiable vector field along \gamma : [0,l] \to S, s \in [0,l] , satisfying the Jacobi equation with J(0) = 0, then J(s) is a Jacobi field along \gamma.

(Def 2) Let \gamma : [0,l] \to S be a geodesic of S with \gamma(0) = p. We say that the point q = \gamma(s_{0}) , s_{0} \in [0,l] , is conjugate to p relative to the geodesic \gamma (if there exists a Jacobi field J(s) which is not identically zero along \gamma with J(0) = J(s_{0}) = J(s_{0}) = 0.

(Prop 3) Let J_{1}(s) and J_{2}(s) be the Jacobi fields along \gamma : [0,l] \to S, s \in [0,l] . Then \langle \frac{DJ_{1}}{ds}, J_{2}(s) \rangle - \langle J_{1}(s) , \frac{DJ_{2}}{ds} \rangle = const.

(Cor) Let J(s) be a Jacobi field along \gamma : [0,l] \to S, with J(0) = J(l) = 0. Then \langle J(s) , \gamma’(s) \rangle = 0, s \in [0,l].

(Prop 5) Let p , q \in S be two points of S and let \gamma : [0,l] \to S be a geodesic joining p = \gamma(0) to q = exp_{p}(l\gamma’(0)) . Then q is conjugate to p relative to \gamma iff v = l\gamma’(0) is a critical point of exp_{p} : T_{p}(S) \to S.

(Thm) Assume that the Gaussian curvature K of a surface S satisfies the condition K \le 0. Then, for every p \in S. the conjugate locus of p is empty. In short, a surface of curvature K \le 0 does not have conjugate points.

(Cor) Assume the Gaussian curvature K of S to be negative or zero. Then for every p \in S, the mapping exp_{p} : T_{p}(S) \to S is a local diffeomorphism.

(Gauss) (Lem 2) Let p \in S be a point of a (complete) surface S and let u \in T_{p}(S) and w \in (T_{p}(S)){u} . Then \langle u, w \rangle = \langle (d exp{p}){u}(u) , (d exp{p}){u}(w) \rangle , where the identification T{p}(S) \approx (T_{p}(S))_{u} is being used.

(subsection 5.6.) Covering Spaces ; The Theorem of Hadamard

(subsubsection A) Covering Spaces

(Def 1) Let \tilde{B} and B be subsets of R^{3} . We say that \pi : \tilde{B} \to B is a covering map if \pi is continuous and \pi(\tilde{B}) = B.

Each point p \in B has a neighborhood U in B (to be called a distinguished neighborhood of p) s.t. \pi^{-1}(U) = \bigcup_{\alpha}V_{\alpha}, where the V_{\alpha}’s are pairwise disjoint open sets s.t. the restriction of \pi to V_{\pi} is a homeomorphism of V_{\alpha} onto U. \tilde{B} is then called a covering space of B.

(Prop 1) Let \pi : \tilde{B} \to B be a local homeomorphism, \tilde{B} compact and B connected. Then \pi is a covering map.

(Prop 2) Let \pi : \tilde{B} \to B be a coveing map, \alpha : [0,l] \to B an arc in B and \tilde{p}{0} \in \tilde{B} a point of \tilde{B} s.t. \pi(\tilde{p}{0}) = \alpha(0) = p_{0} . Then there exists a unique lifting \tilde{\alpha} : [0,l] \to \tilde{B} with origin at \tilde{p}{0} , that is, with \tilde{\alpha}(0) = \tilde{p}{0}.

(Def 2) Let B \subset R^{3} and let \alpha_{0} : [0,l] \to B, \alpha_{1} : [0,l] \to B be two arcs of B, joining the points p = \alpha_{0}(0) = \alpha_{1}(0) , q = \alpha_{0}(l)= \alpha_{1}(l). We say that \alpha_{0} and \alpha_{1} are homotopic if there exists a continuous map H : [0,l] \times [0,l] \to B s.t. H(s,0) = \alpha_{0}(s), H(s,1) = \alpha_{1}(s), s \in [0,l] . H(0,t) = p, H(l,t) = q, t \in [0,1]. The map H is called a homotopy between \alpha_{0} and \alpha_{1}.

(Def) Let \pi : \tilde{B} \to B be a continuous map and let \alpha_{0} , \alpha_{1} \to B be two arcs of B joining the points p and q. Let H : [0,l] \times [0,1] \to B be a homotopy between \alpha_{0} and \alpha_{1}. If there exists a continuous map \tilde{H} : [0,l] \times [0,1] \to \tilde{B} s.t. \pi \bullet \tilde{H} = H, we say that \tilde{H} is a lifting of the homotopy H, with origin at \tilde{H}(0,0) = \tilde{p} \in \tilde{B}.

(prop 3) Let \pi : \tilde{B} \to B be a local homeomorphism with the property of lifting arcs. Let \alpha_{0} , \alpha_{1} : [0,l] \to B be two arcs of B joining the points p and q. Let H : [0,l] \times [0,1] \to B be a homotopy between \alpha_{0} and \alpha_{1} , and let \tilde{p} \in \tilde{B} be a point of \tilde{B} s.t. \pi(\tilde{p}) = p. Then there exists a unique lifting \tilde{H} of H with origin at \tilde{p}.

(Prop 4) Let \pi : \tilde{B} \to B be a local homeomorphism with the property of lifting arcs. Let \alpha_{0} , \alpha_{1} : [0,l] \to B be two arcs of B joining the points p and q and choose \tilde{p} \in \tilde{B} s.t. \pi(\tilde{p}) = p. If \alpha_{0} and \alpha_{1} are homotopic, then the liftings \tilde{\alpha_{0}} and \tilde{\alpha_{1}} , of \alpha_{0} and \alpha_{1}, respectively, with origin \tilde{p} , are homotopic.

(Def 3) An arcwise connected set B \subset R^{3} is simply connected if given two points p, q \in B and two arcs \alpha_{0} : [0,l] \to B, \alpha_{1} : [0,l] \to B joining p to q, there exists a homotopy in B between \alpha_{0} and \alpha_{1} . In particular, any closed arc of B, \alpha:[0,l] \to B (closed means that \alpha(0) = \alpha(l) = p) , is homotopic to the constant arc \alpha(s) = p, s \in [0,l]

(Prop 5) Let \pi : \tilde{B} \to B be a local homeomorphism with the property of lifting arcs. Let \tilde{B} be arcwise connected and B simply connected. Then \pi is a homeomorphism.

(Cor) Let \pi : \tilde{B} \to B be a covering map, \tilde{B} arcwise connected, and B simply connected. Then \pi is a homeomorphism.

(Def 4) B is locally simply connected if any neighborhood of each point contains a simply connected neighborhood.

(Prop 6) Let \pi : \tilde{B} \to B be a local homeomorphism with the property of lifting arcs. Assume that B is locally simply connected and that \tilde{B} is locally arcwise connected. Then \pi is a covering map.

(subsubsection B) The Hadamard Theorem

(Lem 1) Let S be a complete surface of curvature K \le 0. Then exp_{p} : T_{p} (S) \to S , p \in S, is length-increasing in the following sense : If u, w \in T_{p}(S), we have \langle(d exp_{p}){u}(w), (d exp{p}){u}(w) \rangle \ge \langle w, w \rangle, where, as usual, w denotes a vector in (T{p}(S))_{u} that is obtained from w by the translation u.

(Cor) Let K \equiv 0. Then exp_{p} : T_{p}(S) \to S, p \in S, is a local isometry.

(Prop 7) Let S be a complete surface with Gaussian curvature K \le 0. Then the map exp_{p} : T_{p}(S) \to S, p \in S, is a covering map.

(Hadamard)(Thm 1) Let S be a simply connected, complete surface, with Gaussian curvature K \le 0. Then exp_{p} : T_{p}(S) \to S, p \in S, is a diffeomorphism; that is , S is diffeomorphic to a plane.

(Hadamard) (Thm 2) Let S be an ovaloid(Connected, compact, regular surface with Gaussian curvature K > 0) . Then the Gauss map N : S \to S^{2} is a diffeomorphism. In particular, S is diffeomorphic to a sphere.

(subsection 5.7.) Global Theorems for curves; The Fary-Milnor Theorem

(Def) Let \phi : S^{1} \to S^{1} be a continuous map. \phi can be though as continuous map \phi : [0,l] \to S^{1}, with \phi(0) = \phi(l) = p \in S^{1}. Thus, \phi is a closed arc at p in S^{1} which, by (Prop 2) of (Sec 5-6) , can be lifted into a unique arc \tilde{\phi} : [0,l] \to R, starting at a point x \in R with \pi(x) = p. Since \pi(\tilde{\phi}(0)) = \pi(\tilde{\phi}), the difference \tilde{\phi}(l) - \tilde{\phi}(0) is an integral multiple of 2 \pi. The integer given by \tilde{\phi}(l) - \tilde{\phi}(0) = (deg \phi) 2\pi is called the degree of \phi.

(Differentiable Jordan Curve Theorem) (Thm 1) Let \alpha : [0,l] \to R^{2} be a plane, regular, closed, simple curve. Then R^{2} - \alpha([0,l]) has exactly two connected components, and \alpha([0,l]) is their common boundary.

(Def) A plane, regular, closed curve \alpha : [0,l] \to R^{2} is convex if, for each t \in [0,l] , the curve lies in one of the closed half-planes determined by the tangent line at t.

(Thm 2) Let \beta : [0,l] \to R^{2} be a plane, regular, simple, closed curve. Then the rotation index of \beta is \mp 1 (depending on the orientation of \beta)

(Prop 1) A plane, regular, closed curve is convex iff it is simple and its curvature k does not change sign.

(Fenchel’s Theorem) (Thm 3) The total curvature of a simple closed curve is \ge 2\pi, and equality holds iff the curve is a plane convex curve.

(Def) A simple closed continuous curve C \subset R^{3} is unknotted if there exists a homotopy H : S^{1} \times I \to R^{3} , I = [0,1], s.t. H(S^{1} \times {0}) = S^{1}, H(S^{1} \times {1}) = C; and H(S^{1} \times {t}) = C_{t} \subset R^{3} is homeomorphic to S^{1} for all t \in [0,1]. Such a homotopy is called an isotopy; an unknotted curve is then a curve isotopic to S^{1}. When this is not the case, C is said to be knotted.

(Fary-Milnor)(Thm 4) The total curvature of a knotted simple closed curve is greater than 4\pi.

(subsection 5.8.) Surfaces of Zero Gaussian Curvature

(Def) A cylinder is a regular surface S s.t. through each point p \in S there passes a unique line R(p) \subset S (the generator through p) which satisfies the condition that if q \neq p, then the lines R(p) and R(q) are parallel or equal.

(Thm) Let S \subset R^{3} be a complete surface with zero Gaussian curvature. Then S is a cylinder or a plane.

(Prop 1) The unique asymptotic line that passes through a parabolic point p \in U \subset S of a surface S of curvature K \equiv 0 is an (open) segment of a (straight) line in S.

(Massey)(Prop 2) Let r be a maximal asymptotic line passing through a parabolic point p \in U \subset S of a surface S of curvature K \equiv 0 and let P \subset S be the set of planar points of S. Then r \cap P = \empty.

(Lem 1) Let s be the arc length of the asymptotic curve passing through a parabolic point p of a surface S of zero curvature and let H = H(s) be the mean curvature of S along this curve. Then , in U, \frac{d^{2}}{ds^{2}}(\frac{1}{H}) = 0.

(Def) Let now Bd(U) be the boundary of U in S ; that is, Bd(U) is the set of points p \in S s.t. every neighborhood of p in S contains points of U and points of S – U = P.

(Massey)(Prop 3) Let p \in Bd(U) \subset S be a point of the boundary of the set U of parabolic points of a surface S of curvature K \equiv 0. Then through p there passes a unique open segment of line C(p) \subset S . Furthermore, C(p) \subset Bd(U); that is, the boundary of U is formed by segments of lines.

(subsection 5.9.) Jacobi’s Theorems

(Lem 1) Let p \in S, u \in T_{p}(S), l = |u| \neq 0, and let \bar{\gamma} : [0,l] \to T_{p}(S) be the line of T_{p}(S) given by \bar{\gamma}(s) = sv, s \in [0,l] , v = \frac{u}{|u|}. Let \bar{\alpha} : [0,l] \to T_{p}(S) be a differentiable parametrized curve of T_{p}(S), with \bar{\alpha} (0) = 0, \bar{\alpha}(l) = u, and \bar{\alpha}(s) \neq 0 if s \neq 0. Furthermore, let \alpha(s) = exp_{p}(\bar{\alpha})(s) and \gamma(s) = exp_{p}(\bar{\gamma}(s)). We have

l(\alpha) \ge l(\gamma) , where l() denotes the arc length of the corresponding curve.

In addition, if \bar{\alpha}(s) is not a critical point of exp_{p}, s \in [0,l)] , and if the traces of \alpha and \gamma are distinct, then l(\alpha) > l(\gamma).

(Jacobi) (Thm 1) Let \gamma : [0,l] \to S, \gamma(0) = p be a geodesic without conjugate points; that is, exp_{p}: T_{p}(S) \to S is regular at the points of the line \bar{\gamma}(s) = s\gamma’(0) of T_{p}(S), s \in [0,l] . Let h : [0,l] \times (-\epsilon, \epsilon) \to S be a proper variation of \gamma. Then

There exists a \delta >0 , \delta \le \epsilon, s.t. if t \in (-\delta, \delta), L(t) \ge L(0), where L(t) is the length of the curve h_{t} : [0,l] \to S that is given by h_{t}(s) = h(s,t).

If, in addition, the trace of h, is distinct from the trace of \gamma, L(t) > L(0).

(Def 1) Let \gamma : [0,l] \to S be a geodesic of S and let h : [0,l] \times (-\epsilon, \epsilon) \to S be a continuous map with h(s,0) = \gamma(s) , s \in [0,l] . h is said to be a broken variation of \gamma if there exists a partition 0 = s_{0} < s_{1} < s_{2} < \cdots < s_{n-1} < s_{n} = l of [0,l] s.t. h: [s_{i}, s_{i+1}] \times (-\epsilon, \epsilon) \to S, i = 0,1,…, n-1, is differentiable. The broken variation is said to be proper if h(0,t) = \gamma(0) , h(l,t) = \gamma(l) for every t \in (-\epsilon, \epsilon).

(Lem 2) Let V \in \mathcal{U} be a Jacobi Field along a geodesic \gamma : [0,l] \to S and W \to \mathcal{U}. Then I(V,W) = \langle \frac{DV}{ds}(l) , W(l) \rangle - \langle \frac{DV}{ds} (0), W(0) \rangle.

(Jacobi)(Thm 2) If we let \gamma : [0,l] \to S be a geodesic of S and we let \gamma(s_{0}) \in \gamma((0,l)) be a point conjugate to \gamma(0) = p relative to \gamma, then there exists a proper broken variation h : [0,l] \times (-\epsilon, \epsilon) \to S of \gamma and a real number \delta >0, \delta \le \epsilon, s.t. if t \in (-\delta, \delta) we have L(t) \le L(0).

(subsection 5.10.) Abstract Surfaces; Further Generalizations

(Def 1) An abstract surface(differentiable manifold of dimension 2) is a set S together with a family of one-to-one maps x_{\alpha} : U_{\alpha \to S of open sets U_{\alpha} \subset R^{2} into S s.t.

\bigcup_{\alpha}(x_{\alpha})(U_{\alpha}) = S.

For each pair \alpha , \beta with x_{\alpha} (U_{\alpha}) \cap x_{\beta}(U_{\beta}) = W \neq \empty we have that x^{-1}{\alpha}(W) , x{\beta}^{-1} (W) are open sets in R^{2} , and x_{\beta}^{-1} \bullet x_{\alpha}, x_{\alpha}^{-1} \bullet x_{\beta} are differentiable maps.

The pair (U_{\alpha}, x_{\alpha}) with p \in x_{\alpha}(U_{\alpha}) is called a parametrization (or coordinate system) of S around p. x_{\alpha}(U_{\alpha}) is called a coordinate neighborhood, and if q = x_{\alpha} (u_{\alpha}, v_{\alpha}) \in S, we say that (u_{\alpha}, v_{\alpha}) are coordinates of q in this coordinate system. The family {U_{\alpha} , x_{\alpha}} is called a differentiable structure for S.

(Def 2) Let S_{1} and S_{2} be abstract surfaces. A map \phi : S_{1} \to S_{2} is differentiable at p \in S_{1} if given a parametrization y : V \subset R^{2} \to S_{2} around \phi(p) there exists a parametrization x : U \subset R^{2} \to S_{1} around p s.t. \phi(x(U)) \subset y(V) and the map y^{-1} \bullet \phi \bullet x : x^{-1}(U) \subset R^{2} \to R^{2} is differentiable at x^{-1}(p) . \phi is differentiable on S_{1} if it is differentiable at every p \in S_{1} .

(Def 3) A differentiable map \alpha : (-\epsilon, \epsilon) \to S is called a curve on S. Assume that \alpha(0) = p and let D be the set of functions on S which are differentiable at p. The tangent vector to the curve \alpha at t = 0 is the function \alpha’(0) : D \to R given by \alpha’(0) (f) = \frac{d(f \bullet \alpha)}{dt}|_{t = 0} , f \in D. A tangent vector at a point p \in S is the tangent vector at t = 0 of some curve \alpha : (-\epsilon , \epsilon) \to S with \alpha(0) = p.

(Def 4) Let S_{1} and S_{2} be abstract surfaces and let \phi : S_{1} \to S_{2} be a differentiable map. For each p \in S_{1} and each w \in T_{p}(S_{1}) , consider a differentiable curve \alpha : (-\epsilon, \epsilon) \to S_{1} , with \alpha(0) = p, \alpha’(0) = w. Set \beta = \phi \bullet \alpha. The map d\phi_{p} : T_{p} (S_{1}) \to T_{p}(S_{2}) given by d\phi_{p}(w) = \beta’(0) is a well-defined linear map, called the differential of \phi at p.

(Def 5) A geometric surface (Riemannian manifold of dimension 2) is an abstract surface S together with the choice of an inner product \langle , \rangle_{p} at each T_{p} (S), p \in S, which varies differentiably with p in the following sense. For some(and hence all) parametrization x : U \to S around p, the funcitons E(u,v) = \langle \frac{ \partial }{\partial u } , \frac{\partial }{\partial u } \rangle , F(u,v) = \langle \frac{\partial }{\partial u} , \frac{\partial }{\partial v} \rangle , G(u,v) = \langle \frac{\partial }{\partial v} , \frac{\partial }{\partial v} \rangle are differentiable functions in U. The inner product \langle , \rangle is often called a (Riemannian) metric on S.

(Def 6) A differentiable map \phi : S \to R^{3} of an abstract surface S into R^{3} is an immersion if the differential d \phi_{p} : T_{p} (S) \to T_{p}(R^{3}) is injective. If, in addition, S has a metric \langle, \rangle and \langle d \phi_{p}(v), d \phi_{p}(w) \rangle_{\phi(p)} = \langle v , w \rangle_{p} , v , w \in T_{p}(S), \phi is said to be an isometric immersion.

(Def 7) Let S be an abstract surface. A differentiable map \phi : S \to R^{n} is an embedding if \phi is an immersion and a homeomorphism onto its image.

(Def 1a) A differentiable manifold of dimension n is a set M together with a family of one-to-one maps x_{\alpha} : U_{\alpha} \to M of open sets U_{\alpha} \subset R^{n} into M s.t.

\bigcup_{\alpha}(x_{\alpha})(U_{\alpha}) = M.

For each pair \alpha , \beta with x_{\alpha} (U_{\alpha}) \cap x_{\beta}(U_{\beta}) = W \neq \empty we have that x^{-1}{\alpha}(W) , x{\beta}^{-1} (W) are open sets in R^{n} , and x_{\beta}^{-1} \bullet x_{\alpha}, x_{\alpha}^{-1} \bullet x_{\beta} are differentiable maps.

The family {U_{\alpha}, x_{\alpha}} is maximal relative to first and second conditions.

The family {U_{\alpha}, x_{\alpha}} satisfying first and second conditions is called a differentiable structure on M.

(Def 5a) ) A Riemannian manifold is an n-dimensional differentiable manifold M together with a choice, for each p \in M, of an inner product \langle , \rangle_{p} at each T_{p} (M), p \in M, which varies differentiably with p in the following sense. For some parametrization x_{\alpha} : U_{\alpha} \to M with p \in x_{\alpha}(U_{\alpha}) , the functions g_{ij}(u_{1},…,u_{n}) = \langle \frac{\partial}{\partial u_{i}}, \frac{\partial}{\partial u_{j}} \rangle , i, j = 1, …, n , are differentiable at x_{\alpha}^{-1} (p) ; here (u_{1}, …, u_{n}) are the coordinates of U_{\alpha} \subset R^{n}.

(subsection 5.11) Hilbert’s Theorem

(Thm) A complete geometric surface S with constant negative curvature cannot be isometrically immersed in R^{3}.

(Def) assume the curvature K \equiv -1. Since exp_{p} : T_{p}(S) \to S is a local diffeomorphism , it induces an inner product in T_{p}(S). Denote by S’ the geometric surface T_{p}(S) with this inner product.

(Lem 1) The area of S’ is infinite.

(Def) Assume there exists an isometric immersion \phi : S’ \to R^{3} , where S’ is a geometric surface homeomorphic to a plane and with K \equiv -1.

(Lem 2) For each p \in S’ there is a parametrization x : U \subset R^{2} \to S’, p \in x(U) , s.t. the coordinate curves of x are the asymptotic curves of x(U) = V’ and form a Tchbyshef net. (we shall express this by saying that the asymptotic curves of V’ form a Tchebyshef net.)

(Lem 3) Let V’ \subset S’ be a coordinate neighborhood of S’ s.t. the coordinate curves are the asymptotic curves in V’. Then the area A of any quadrilateral formed by the coordinate curves is smaller than 2\pi.

(Lem 4) For a fixed t, the curve x(s,t) , -\infty < s < \infty , is an asymptotic curve with s as arc length.

(Lem 5) x is a local diffeomorphism.

(Lem 6) x is surjective.

(Lem 7) On S’ there are two differentiable linearly independent vector fields which are tangent to the asymptotic curves of S’.

(Lem 8) x is injective.

(Appendix) Point-Set Topology of Euclidean Spaces

(subsection A) Preliminaries

(Def 1) A sequence p_{1}, …, p_{i}, …, \in R^{n} converges to p_{0} \in R^{n} if given \epsilon >0, there exists an index i_{0} of the sequence s.t. p_{i} \in B_{e}(p_{0}) for all i > i_{0} . In this situation, p_{0} is the limit of the sequence {p_{i}} and this is denoted by {p_{i}} \to p_{0}.

(Prop 1) A map F : U \subset R^{n} \to R^{m} is continuous at p_{0} \in U iff for each converging sequence {p_{i}} \to p_{0} in U, the sequence {F(p_{i})} converges to F(p_{0}).

(Def 2) A point p \in R^{n} is a limit point of a set A \subset R^{n} if every neighborhood of p in R^{n} contains one point of A distinct from p.

(Def 3) A set F \subset R^{n} is closed if every limit point of F belongs to F. The closure of A \subset R^{n} denoted by \bar{A} is the union of A with its limit points.

(Prop 2) F \subset R^{n} is closed iff the complement R^{n} – F of F is open.

(Prop 3) A map F : U \subset R^{n} \to R^{m} is continuous iff for each open set V \subset R^{m} , F^{-1}(V) is an open set.

(Cor) F : U \subset R^{n} \to R^{m} is continuous iff for every closed set A \subset R^{m}, F^{-1} (A) is a closed set.

(Def 4) Let A \subset R^{n}. The boundary Bd A of A is the set of points p in R^{n} s.t. every neighborhood of p contains points in A and points in R^{n} – A.

(Def 5) Let A \subset R^{n} . We say that V \subset A is an open set in A if there exists an open set U in R^{n} s.t. V = U \cap A. A neighborhood of p \in A in A is an open set in A containing p.

(Def 6) A subset A \subset R of the real line R is bounded above if there exists M \in R s.t. M \ge a for all a \in A. The number M is called an upper bound for A. When A is bounded above, a supremum or a least upper bound of A, sup A(or l.u.b. A) is an upper bound M which satisfies the following condition : Given \epsilon >0, there exists a \in A s.t. M - \epsilon < a. By changing the sign of the above inequalities, we define similarly a lower bound for A and an infimum (or a greatest lower bound) of A, inf A (or g.l.b. A).

(Axiom of Completeness of Real numbers) Let A \subset R be nonempty and bounded above(below). Then there exists sup A (inf A).

(Lem 1) Call a sequence {x_{i}} of real numbers a Cauchy sequence if given \epsilon <0, there exists i_{0} s.t. |x_{i} – x_{j}| < \epsilon for all i,j > i_{0}. A sequence is convergent iff it is a Cauchy sequence.

(Def 7) A sequence {p_{i}} , p_{i} \in R^{n}, is a Cauchy sequence if given \epsilon >0, there exists an index i_{0} s.t. the distance |p_{i} – p_{j}| < \epsilon for all i,j > i_{0}.

(Prop 4) A sequence {p_{i}} , p_{i} \in R^{n}, converges iff it is a Cauchy sequence.

(subsection B) Connected sets

(Def 8) A continuous curve \alpha : [a,b] \to A \subset R^{n} is called an arc in A joining \alpha(a) to \alpha(b).

(Def 9) A \subset R^{n} is arcwise connected if , given two points p, q \in A, there exists an arc in A joining p to q.

(Def 10) A \subset R^{n} is connected when it is not possible to write A = U_{1} \cap U_{2} , where U_{1} and U_{2} are nonempty open sets in A and U_{1} \cap U_{2} = \empty.

(Prop 5) Let A \subset R^{n} be connected and let B \subset A be simultaneously open and closed in A. Then either B = \empty or B = A.

(Prop 6) Let F : A \subset R^{n} \to R^{m} be continuous and A be connected. Then F(A) is connected.

(Def 11) An interval of the real line R is any of the sets a < x < b, a \le x \le b, a < x \le b , a \le x < b , x \in R, The cases a = b , a = - \infty , b = + \infty are not excluded, so that an interval may be a point , a half-line, or R itself.

(Prop 7) A \subset R is connected iff A is an interval.

(Prop 8) Let f : A \subset R^{n} \to R be continuous and A be connected. Assume that f(q) \neq 0 for all q \in A. Then f does not change sign in A.

(Prop 9) Let A \subset R^{n} be arcwise connected. Then A is connected.

(Def 12) A set A \subset R^{n} is locally arcwise connected if for each p \in A and each neighborhood V of p in A there exists an arcwise connected neighborhood U \subset V of p in A.

(Prop 10) Let A \subset R^{n} be a locally arcwise connected set. Then A is connected iff it is arcwise connected.

(subsection C) Compact sets

(Def 13) A set A \subset R^{n} is bounded if it is contained in some ball of R^{n}. A set K \subset R^{n} is compact if it is closed and bounded.

(Def 14) An open cover of a set A \subset R^{n} is a family of open sets {U_{\alpha}} , \alpha \in \mathfrak{A} s.t. \bigcup_{\alpha} U_{\alpha} = A. When there are only finitely many U_{\alpha} in the family, we say that the cover is finite. If the subfamily {U_{\beta}} , \beta \in \mathfrak{B} \subset \mathfrak{A} , still covers A, that is, \bigcup_{\beta} U_{\beta{ = A, we say that {U_{\beta}} is a subcover of {U_{\alpha}} .

(Prop 11) For a set K \subset R^{n} the following assertions are equivalent.

K is compact.

(Heine-Borel) Every open cover of K has a finite subcover.

(Bolzano-Weierstrass) Every infinite subset of K has a limit point in K.

(Prop 12) Let F : K \subset R^{n} \to R^{m} be continuous and let K be compact. Then F(K) is compact.

(Prop 13) Let f : K \subset R^{n} \to R be a continuous function defined on a compact set K. Then there exists p_{1}, p_{2} \in K s.t. f(p_{2}) \le f(p) \le f(p_{1}) for all p \in K; that is, f reaches a maximum at p_{1} and a minimum at p_{2}.

(subsection D) Connected Components

(Prop 14) Let C_{\alpha} \subset R^{n} be a family of connected sets s.t. \bigcap_{\alpha} C_{\alpha} \neq \empty. Then \bigcup_{\alpha} C_{\alpha} = C is a connected set.

(Def 15) Let A \subset R^{n} and p \in A. The union of all connected subsets of A which contain p is called the connected component of A containing p.

(Prop 15) Let C \subset A \subset R^{n} be a connected set. Then the closure \bar{C} of C in A is connected.

(Cor) A connected component C \subset A \subset R^{n} of a set A is closed in A.

(prop 16) Let C \subset A \subset R^{n} be a connected component of a locally arcwise connected set A. Then C is open in A.

Topology

Fundamental concepts of Topology

1 Set theory and Logic

1.1 Fundamental concepts

1.1.1 Basic Notation

1.1.1.1 Sets <- Capital letters

1.1.1.2 Objects (elements) <- lowercase letters

1.1.1.3 Logical Identity <- = (equality symbol)

1.1.1.4 A ⊂ B <- A is a subset of B , inclusion

1.1.1.4.1 Every element of A is also an element of B

1.1.1.5 A \subsetneq B <- A is a proper subset of B, proper inclusion

1.1.1.5.1 A \subset B and A is different from B

1.1.2 The Union of Sets and the meaning of ‘or’

1.1.2.1 Union of A and B ( A \cup B) = { x | x \in A or x \in B}

1.1.2.1.1 P or Q : P or Q or both

1.1.3 The Intersection of Sets, the Empty Set and the Meaning of “if .. Then”

1.1.3.1 Intersection of A and B ( A \cap B) = { x | x \in A and x \in B}

1.1.3.2 Empty set \empty : the set having no elements

1.1.3.2.1 A \cup \empty = A

1.1.3.2.2 A \cap \empty = \empty

1.1.3.3 Disjoint (a set A, a set B) : A \cap B = \empty

1.1.3.4 If (Hypothesis), then (conclusion)

1.1.3.4.1 Vacuously true : no case for which the hypothesis holds

1.1.4 Contrapositive and Converse

1.1.4.1 Contrapositive ( ‘If P, then Q ‘ statement) : If Q is not true, then P is not true

1.1.4.2 Converse ( ‘If P, then Q ‘ statement) : If Q, then P

1.1.4.3 P <-> Q : P holds iff Q holds

1.1.5 Negation ( a statement P ) : not P

1.1.6 The difference of two sets

1.1.6.1 Difference of two sets A, B ( Complement of B relative to A)

1.1.6.1.1 A – B = { x | x \in A and x \notin B}

1.1.7 Rules of set theory

1.1.7.1 Distributive law (sets A, B, C)

1.1.7.1.1 A \cap (B \cup C) = (A \cap B) \cup (A \cap C)

1.1.7.1.2 A \cup (B \cap C) = (A \cup B) \cap (A \cup C)

1.1.7.2 Demorgan’s laws

1.1.7.2.1 A – (B \cup C) = (A – B) \cap (A – C)

1.1.7.2.2 A – (B \cap C) = (A – B) \cup (A – C)

1.1.8 Collection of the sets

1.1.8.1 Power set of A ( \mathcal{P} (A) ): set of all subsets of A

1.1.8.2 Collection of sets : a set whose elements are sets

1.1.9 Arbitrary unions and intersections

1.1.9.1 Given a collection \mathcal{A} of sets,

1.1.9.1.1 Union of the elements of \mathcal{A}

1.1.9.1.1.1 \bigcup_{A \in \mathcal{A}} A = { x | x \in A for at least one A \in \mathcal{A}}

1.1.9.1.2 Intersection of the elements of \mathcal{A}

1.1.9.1.2.1 \bigcap_{A \in \mathcal{A}} A = { x | x \in A for every A \in \mathcal{A}}

1.1.9.1.3 \bigcup_{A \in \mathcal{A}} A = \empty

1.1.9.1.4 \bigcap_{A \in \mathcal{A}} A = (Not defined)

1.1.10 Cartesian Product

1.1.10.1 Ordered pair of objects a , b : a \times b

1.1.10.1.1 All ordered pairs of real numbers is a plane

1.1.10.2 Cartesian product ( sets A , B) : A \times B = {a \times b | a \in A and b \in B}

1.2 Function

1.2.1 A rule of assignment : a subset r of the cartesian product C x D of two sets, having the property that each element of C appears as the first coordinate of at most one ordered pair belonging to r.

1.2.1.1 [c \times d \in r and c \times d’ \in r] -> [d = d’]

1.2.1.2 Domain (a rule of assignment r) : { c | there exists d \in D s.t. c \times d \in r}

1.2.1.3 Image ( a rule of assignment r) : {d | there exists c \in C s.t. c \times d \in r}

1.2.2 Function f : a rule of assignment r, together with a set B that contains the image set of r.

1.2.2.1 Domain (a function f) : Domain of the rule r

1.2.2.2 Image set (a function f) : the image set of r

1.2.2.3 Range ( a function f) : the set B

1.2.2.3.1 f is a function having domain A and range B : f : A -> B

1.2.2.4 value ( function f, element a of Domain (f) ) : unique element of B the rule determining f assigns to a

1.2.3 Restriction of function f | A_{0} : If f : A -> B and if A_{0} is a subset of A, we define the restriction of f to A_{0} to be the function mapping A_{0} into B whose rule is {(a,f(a)) | a \in A_{0}}

1.2.4 Composite g \bullet f : given functions f : A->B and g : B->C, the composite of f and g is the function g \bullet f : A -> C defined by equation (g \bullet f)(a) = g(f(a))

1.2.5 Injective : A function f : A -> B is said to be injective (or one -to- one) if for each pair of distinct points of A, their images under f are distinct.

1.2.6 Surjective : f is said to map A onto B if every element of B is the image of some element of A under the function f.

1.2.7 Bijective : f is said to be one-to-one correspondence if f is both injective and surjective

1.2.7.1 If f is bijective, there exists a function from B to A, called the inverse of f. f^{-1}

1.2.7.2 (Criterion of bijection) : Let f : A->B. if there are functions g : B->A and h:B->A s.t. g(f(a)) = a for every a in A and f(h(b)) = b for every b in B, then f is bijective and g = h = f^{-1}

1.2.8 Image of A_{0} under f : Let f:A->B. if A_{0} is a subset of A, we denote by f(A_{0}) the set of all images of points of A_{0} under the function.

1.2.8.1 Preimage of B_{0} under f : if B_{0} is a subset of B, f^{-1}(B_{0}) is the set of all elements of A whose images under f lie in B_{0}.

1.2.8.2 A_{0} \subseteq f^{-1}(f(A_{0})) (equality if f is injective)

1.2.8.3 F(f^{-1}(B_{0})) \subseteq B_{0} (equality if f is surjective)

1.3 Relations

1.3.1 A relation on a set A : a subset C of the cartesian product A \times A

1.3.2 Equivalence relation on a set A : a relation C on A with

1.3.2.1 (Reflexivity) : xCx for every x in A

1.3.2.2 (Symmetry) if xCy, then yCx

1.3.2.3 (Transitivity) if xCy and yCz, then xCz

1.3.3 Equivalence class determined by x : given an equivalence relation ~ on a set A and an element x of A, we define a certain subset E of A, by the equation E = { y | y ~ x }

1.3.3.1 Two equivalence classes E and E’ are either disjoint or equal.

1.3.4 Partition of a set A : a collection of disjoint nonempty subsets of A whose union is all of A.

1.3.4.1 Given any partition \mathcal{D} of A, there is exactly one equivalence relation on A from which it is derived.

1.3.5 Order relations ( a relation C on a set A)

1.3.5.1 (Comparability) For every x and y in A for which x \neq y, either xCy or yCx.

1.3.5.2 (nonreflexivity) : For no x in A does the relation xCx hold.

1.3.5.3 (Transitivity) If xCy and yCz, then xCz

1.3.6 Open interval in X : if X is a set and < is an order relation on X, and if a < b, we use the notation (a, b) to denote the set {x | a < x < b }

1.3.6.1 a is the immediate predecessor of b, and b is the immediate predecessor of a if (a, b) is empty

1.3.7 Suppose that A and B are two sets with order relations <{A} and <{B} respectively, We say that A and B have the same order type if there is a bijective correspondence between them that preserves order

1.3.7.1 Preserve order : a_{1} <{A} a{2} -> f(a_{1}) <{B} f(a{2})

1.3.8 Dictionary order relation : Suppose A and B are two sets with order relations <{A} and <{B} respectively, define an order relation on A \times B by defining a_{1} \times b_{1} < a_{2} times b_{2} if a_{1}<{A}a{2} or if a_{1} = a_{2} and b_{1} <{B} b{2}.

1.3.9 Largest element of A_{0} (an element b) : Suppose that A is a set ordered by relation <. Let A_{0} be a subset of A. b \in A_{0} and if x \le b for every x \in A_{0}

1.3.10 Smallest element of A_{0} (an element a) : a \in A_{0} and if a \le x for every x \in A_{0}

1.3.11 Bounded above (a subset A_{0} of A) : there is an element b of A s.t. x \le b for every x \in A_{0}.

1.3.11.1 Upper bound for A_{0} : b

1.3.11.2 The least upper bound of A_{0} [supremum of A_{0}, sup A_{0}] : a smallest element of the set of all upper bounds for A_{0}

1.3.11.3 Least upper bound property ( an ordered set A) : every nonempty subset A_{0} of A that is bounded above has a least upper bound

1.3.12 Bounded below (a subset A_{0} of A) : there is an element a of A s.t. a \le x for every x \in A_{0}.

1.3.12.1 Lower bound for A_{0} : a

1.3.12.2 The greatest lower bound of A_{0} [infimum of A_{0}, inf A_{0}] : a largest element of the set of all lower bounds for A_{0}

1.3.12.3 Greatest lower bound property (an ordered set A) : every nonempty subset A_{0} of A that is bounded below has the greatest lower bound property.

1.4 The Integers and the real numbers

1.4.1 Binary operation on a set A : a function f mapping A \times A into A

1.4.2 Assume the existence of the set of real numbers \mathbb{R} with +, \cdot, <

1.4.2.1 Algebraic properties

1.4.2.1.1 (x+y) + z = x + (y+z) , (x \cdot y) \cdot z = x \cdot (y \cdot z) for all x, y, z in \mathbb{R}

1.4.2.1.2 x + y = y + x, x \cdot y = y \cdot x for all x,y in \mathbb{R}

1.4.2.1.3 There exists a unique element of \mathbb{R} called zero, denoted by 0, s.t. x + 0 = x for all x \in \mathbb{R}, There exists a unique element of \mathbb{R} called one, denoted by 1, s.t. x \cdot 1 = x for all x \in \mathbb{R}

1.4.2.1.4 (Negative of x) For each x in \mathbb{R}, there exists a unique y in \mathbb{R} s.t. x + y = 0,

1.4.2.1.5 (Reciprocal of x) For each x in \mathbb{R} different from 0, there exists a unique y in \mathbb{R} s.t. x\cdot y = 1.

1.4.2.1.6 x \cdot (y+z) = (x \cdot y) + (x \cdot z) for all x, y, z \in \mathbb{R}.

1.4.2.2 A Mixed Algebraic and Order Property

1.4.2.2.1 if x > y, then x + z > y + z. If x > y and z > 0, then x \cdot z > y \cdot z

1.4.2.3 Order properties

1.4.2.3.1 The order relation < has the least upper bound property.

1.4.2.3.2 If x < y, there exists an element z s.t. x < z and z < y

1.4.2.4 Subtraction operation : z-x = z + (-x)

1.4.2.5 Quotient z/x = z \cdot (1/x)

1.4.2.6 Laws of inequalities : If x > y and z < 0, then x \cdot z < y \cdot z

1.4.2.7 Field : only algebraic properties of above

1.4.2.8 Ordered field : algebraic property and a mixed algebraic and order property of above

1.4.2.9 Linear continuum : only order properties of above

1.4.3 Inductive ( a subset of \mathbb{R}) : it contains the number 1 and if for every x in A, the number x + 1 is also in A.

1.4.4 Positive Integers Z_{+} = \bigcap_{A \in \mathcal{A}} A

1.4.4.1 Z_{+} is inductive

1.4.4.2 (Principle of Induction) : if A is an inductive set of positive integers, then A = Z_{+}

1.4.5 Integers \mathbb{Z} : a set consisting of Z_{+}, the number 0, the negatives of the elements of Z_{+}

1.4.6 Rational numbers \mathbb{Q} : a set of quotients of integers

1.4.7 Section S_{n} of the positive integers : S_{n+1} = {1,…,n}, S_{1} = \empty

1.4.8 (Well-ordering property) : Every nonempty subset of \mathbb{Z}_{+} has a smallest element.

1.4.9 (Strong induction principle) : Let A be a set of positive integers. Suppose that for each positive integer n, the statement S_{n} \subset A implies the statement n \in A, then A = \mathbb{Z}_{+}.

1.4.10 Proofs using least upper bound axiom

1.4.10.1 Archimedian ordering property of the real line : \mathbb{Z}_{+} of positive integers has no upper bound in \mathbb{R}

1.4.10.2 Greatest lower bound property

1.4.10.3 Existence of a unique positive square root for every positive real number

1.5 Cartesian Products

1.5.1 Let \mathcal{A} be a nonempty collection of sets.

1.5.1.1 Indexing function (\mathcal{A}) : a surjective function f from some set J to \mathcal{A}.

1.5.1.1.1 Index set : J

1.5.1.1.2 Indexed family of sets {A_{\alpha}}_{\alpha \in J} : the collection \mathcal{A} with indexing function f

1.5.1.1.2.1 A_{\alpha} = f(\alpha)

1.5.2 \bigcup_{\alpha \in J} A_{\alpha} = {x | for at least one \alpha \in J, x \in A_{\alpha}}

1.5.3 \bigcap_{\alpha \in J} A_{\alpha} = {x | for every \alpha \in J, x \in A_{\alpha}}

1.5.4 M-tuple of elements of X : Let m be a positive integer. Given a set, a function \mathbf{x} : {1,…,m} -> X

1.5.4.1 (x_{1}, …, x_{m})

1.5.4.2 Ith coordinate of \mathbf{x} : value of \mathbf{x} at I

1.5.4.3 Cartesian product ({A_{1}, …, A_{m}} being a family of sets indexed with the set {1,…,m}) :

1.5.4.3.1 Let X = A_{1} \cup … \cup A_{m}. then \prod_{I = 1}^{m} A_{i} or A_{1} \times … \times A_{m} = set of all m-tuples (x_{1} , …, x_{m}) of elements of X s.t. x_{i} \in A_{i} for each i.

1.5.4.4 X^{m} <- M-tuple of elements of X

1.5.5 \omega -tuple of elements of a set X : \mathbf{x} : \mathbb{Z}_{+} -> X

1.5.5.1 Also called sequence , infinite sequence, (x_{1}, x_{2}, …) , (x_{n}){n \in \mathbb{Z}{+}_

1.5.5.2 Ith coordinate of \mathbf{x} : value of \mathbf{x} at I

1.5.5.3 Cartesian product ({A_{1}, A_{2}, … } being a family of sets indexed with positive integers) :

1.5.5.4 Let X = Union of the sets in this family. then \prod_{I = \mathbb{Z}{+}} A{i} or A_{1} \times A_{2} \times… = set of all \omega -tuples (x_{1} , x_{2}… ) of elements of X s.t. x_{i} \in A_{i} for each i.

1.5.5.5 X^{\omega} <- 1.5.5 \omega -tuple of elements of a set X

1.6 Finite sets

1.6.1 Finite (a set) : there is a bijective correspondence of A with some section of the positive integers.

1.6.1.1 Cardinality 0 : A is empty

1.6.1.2 Cardinality n : there is a bijection f:A->{1,..,n} for some positive integer n

1.6.2 (Thm 6.1) Let A be a set. Suppose that there exists a bijection f:A-> {1,..,n} for some n \in \mathbb{Z}_{+}. Let B be a proper subset of A. then there exists no bijection g : B->{1,…,n}. But provided B \neq \empty, there does exist a bijection h:B->{1,…,m} for some m<n.

1.6.2.1 Let n be a positive integer. Let A be a set. Let a_{0} be an element of A. There exists a bijective correspondence f of the set A with the set {1,…,n+1} iff there exists a bijective correspondence g of the set A-{a_{0}} with the set {1,…,n}.

1.6.2.2 (Cor) If A is finite, there is no bijection of A with a proper subset of itself.

1.6.2.3 (Cor) Z_{+} is not finite.

1.6.2.4 (Cor) The cardinality of a finite set A is uniquely determined by A.

1.6.2.5 (Cor) If B is a subset of the finite set A, then B is finite. If B is a proper subset of A, then the cardinality of B is less than the cardinality of A.

1.6.2.6 (Cor) Let B be a nonempty set. Then the following are equivalent.

1.6.2.6.1 B is finite.

1.6.2.6.2 There is a surjective function from a section of the positive integers onto B.

1.6.2.6.3 There is an injective function from B into a section of the positive integers.

1.6.2.7 (Cor) Finite unions and finite cartesian products of finite sets are finite.

1.6.2.7.1 If A and B are finite, so is A \cup B.

1.7 Countable and Uncountable Sets

1.7.1 Infinite : not finite

1.7.2 Countably infinite : there is a bijective correspondence f : A-> \mathbb{Z}_{+}

1.7.3 Countable (a set) : it is finite or countably finite

1.7.4 Uncountable (a set) : not countable

1.7.5 (Thm 7.1) : Let B be a nonempty set. Then the following are equivalent.

1.7.5.1 B is countable.

1.7.5.2 There is a surjective function f : \mathbb{Z}_{+} -> B

1.7.5.3 There is an injective function g : \B -> \mathbb{Z}_{+}

1.7.5.4 (Lem) If C is an infinite subset of \mathbb{Z}_{+}, then C is countably infinite.

1.7.5.4.1 H(n) = smallest element of [C- H({1,…,n-1})], H(1) = smallest element of C

1.7.6 Principle of recursive definition : Let A be a set. A recursive formula determines a unique function h: \mathbb{Z}_{+} -> A.

1.7.6.1 Recursive formula : a formula that defines h(1) as a unique element of A, and for i>1 defines h(i) uniquely as an element of A in terms of the values of h for positive integers less than I.

1.7.7 (Cor) A subset of a countable set is countable.

1.7.8 (Cor) the set \mathbb{Z}{+} \times \mathbb{Z}{+} is countably infinite.

1.7.9 (Thm 7.5) A countable union of countable sets is countable.

1.7.10 (Thm 7.6) A finite product of countable sets is countable.

1.7.11 (Thm 7.7) Let X be the two element set {0,1}. Then the set X^{\omega} is uncountable.

1.7.12 (Thm 7.8) Let A be a set. There is no injective map f: \mathcal{P} (A) -> A, and there is no surjective map g: A->\mathcal{P}

1.8 The principle of Recursive Definition

1.8.1 (Thm 8.3) There exists a unique function h: \mathbb{Z}{+} -> C satisfying recursive formula for all I \in \mathbb{Z}{+}.

1.8.1.1 (Lem)Given n \in \mathbb{Z}_{+}, there exists a function f:{1,…,n} ->C that satisfies the recursive formula for all I in its domain.

1.8.1.2 (Lem)Suppose that f:{1,…,n} -> C and g : {1,…,m} ->C both satisfy recursive formula for all I in their respective domains. Then f(i) = g(i) for all I in both domains.

1.8.2 (Thm 8.4) (Principle of recursive definition) : let A be a set; let a_{0} be an element of A. Suppose \rho is a function that assigns, to each function f mapping a nonempty section of the positive integers into A, an element of A. Then there exists a unique function h: \mathbb{Z}_{+} -> A s.t. recursion formula for h satisfied.

1.8.2.1 Recursion formula for h :

1.8.2.1.1 H(1) = a_{0}

1.8.2.1.2 H(i) = \rho(h | {1,…,i-1}) for I >1

1.9 Infinite sets and the axiom of choice

1.9.1 (Thm 9.1) Let A be a set. The following statements about A are equivalent.

1.9.1.1 There exists an injective function f : \mathbb{Z}_{+} -> A

1.9.1.2 There exists a bijection of A with a proper subset of itself.

1.9.1.3 A is infinite.

1.9.1.4 Pf) Axiom of choice is needed., recursion formula, c: \mathcal{B} -> \bigcup_{B \in \mathcal{B}} B = A

1.9.2 (Axiom of choice) : Given a collection \mathcal{A} of disjoint nonempty sets, there exists a set C consisting of exactly one element from each element of \mathcal{A}

1.9.2.1 A set C s.t. C is contained in the union of the elements of \mathcal{A}, and for each A \in \mathcal{A}, the set C\cap A contains a single element.

1.9.3 (Existence of a choice function)(Lem)

1.9.3.1 Given a collection \mathcal{B} of nonempty sets not necessarily disjoint, there exists a function c : \mathcal{B} -> \bigcup_{B \in \mathcal{B}} B s.t. c(B) is an element of B, for each B \in \mathcal{B}.

1.9.3.1.1 A function c is called a choice function for the collection \mathcal{B} .

1.9.4 Finite axiom of choice : given a finite collection \mathcal{A} of disjoint nonempty sets, there exists a set \mathcal{C} consisting of exactly one element from each element of \mathcal{A}.

1.9.4.1 Weaker form of axiom of choice.

1.10 Well-Ordered sets

1.10.1 Well-ordered ( a set A with an order relation < ) : every nonempty subset of A has a smallest element.

1.10.1.1 Constructing well-ordered sets

1.10.1.1.1 If A is a well-ordered set, then any subset of A is well-ordered in the restricted order relation.

1.10.1.1.2 If A and B are well-ordered sets, then A \times B is well-ordered in the dictionary order.

1.10.2 (Thm 10.1) Every nonempty finite ordered set has the order type of a section {1,…,n} of \mathbb{Z}_{+}, so it is well-ordered.

1.10.3 (Well-ordering theorem) If A is a set, there exists an order relation on A that is well-ordering.

1.10.3.1 Pf) choice axiom

1.10.4 (Cor) There exists an uncountable well-ordered set.

1.10.5 Section S_{\alpha} of X by \alpha : Let X be a well-ordered set. Given \alpha \in X, S_{\alpha} = {x | x \in X and x < \alpha}

1.10.6 (Thm 10.3) If A is a countable subset of S_{\Omega}, then A has an upper bound in S_{\Omega}.

1.10.6.1 (Lem) There exists a well-ordered set A having a largest element \Omega, s.t. the section S_{\Omega} of A by \Omega is uncountable but every other section of A is countable.

1.11 The Maximum principle

1.11.1 Strict partial order on A (a relation < on A)

1.11.1.1 (Nonreflexivity) The relation a<a never holds

1.11.1.2 (Transitivity) If a<b and b<c, then a<c

1.11.2 (The maximum principle) Let A be a set; let < be a strict partial order on A. Then there exists a maximal simply ordered subset of B.

1.11.2.1 Pf) well-ordering theorem

1.11.3 Let A be a set and let < be a strict partial order on A.

1.11.3.1 Upper bound on B ( B a subset of A) : an element c of A s.t. for every b in B, either b = c or b < c.

1.11.3.2 Maximal element of A : an element m of A s.t. for no element a of A does the relation m < a hold.

1.11.4 (Zorn’s Lemma) : Let A be a set that is strictly partially ordered. If every simply ordered subset of A has an upper bound in A, then A has a maximal element.

1.11.5 Partial order on A : let < be a strict partial order on a set A. Then if we define a \le b either a < b or a = b, then the relation \le is a partial order.

Topological spaces and continuous functions

2 Topological spaces and continuous functions

2.1 Topological spaces

2.1.1 Topology on a set X : a collection \Tau of subsets of X having following properties.

2.1.1.1 \empty and X are in \Tau.

2.1.1.2 Union of the elements of any subcollection of \Tau is in \Tau.

2.1.1.3 Intersection of the elements of any finite subcollection of \Tau is in \Tau.

2.1.2 Topological space : ordered pair (X, \Tau)

2.1.2.1 X <- ordered pair (X, \Tau)

2.1.3 Open set of X (a subset U of X) : Let X is a topological space with topology \Tau, if U belongs to the collection \Tau.

2.1.4 Sort

2.1.4.1 Discrete topology ( a set X) : a collection of all subsets of X

2.1.4.2 Indiscrete topology ( a set X) : {X, \empty} ~ trivial topology

2.1.4.3 Finite complement topology \Tau_{f} ( a set X) : Collection of all subsets U of X s.t. X-U either is finite or is all of X.

2.1.5 Suppose \Tau and \Tau’ are two topologies on a given set X.

2.1.5.1 \Tau’ is finer than \Tau : \Tau \subset \Tau’

2.1.5.2 \Tau’ is strictly finer than \Tau : \Tau \subsetneq \Tau’

2.1.5.3 \Tau’ is coarser than \Tau : \Tau’ \subset \Tau

2.1.5.4 \Tau’ is strictly coarser than \Tau : \Tau’ \subsetneq \Tau

2.1.5.5 \Tau is comparable with \Tau’ : \Tau’ \subset \Tau or \Tau \subset \Tau’

2.2 Basis for a Topology

2.2.1 Basis for a topology on X (a set X) : a collection \mathcal{B} of subsets of X (called basis elements) s.t.

2.2.1.1 For each x \in X, there is at least one basis element B containing x.

2.2.1.2 If x belongs to the intersection of two basis elements B_{1} and B_{2}, then there is a basis element B_{3} containing x s.t. B_{3} \subset B_{1} \cap B_{2}.

2.2.2 Topology \Tau generated by \mathcal{B}

2.2.2.1 Open in X ( a subset U of X) : for each x \in U, there is a basis element B \in \mathcal{B} s.t. x \in B and B \subset U.

2.2.3 (Lem 13.1) : Let X be a set; let \mathcal{B} be a basis for a topology \Tau on X. then \Tau equals the collection of all unions of elements of \mathcal{B}.

2.2.4 (Lem 13.2) : Let X be a topological space. Suppose that \mathcal{C} is a collection of open sets of X s.t. for each open set U of X and each x in U , there is an element C of \mathcal{C} s.t. x \in C \subset U. Then \mathcal{C} is a basis for the topology of X.

2.2.5 (Lem 13.3) Let \mathcal{B} and \mathcal{B} be bases for the topologies \Tau and \Tau’ respectively on X. Then the following are equivalent.

2.2.5.1 \Tau’ is finer than \Tau.

2.2.5.2 For each x \in X and each basis element B \in \mathcal{B} containing x, there is a basis element B’ \in \mathcal{B}’ s.t. x \in B’ \subset B.

2.2.6 Standard topology on the real line : Topology generated by \mathcal{B}

2.2.6.1 \mathcal{B} is the collection of all open intervals in the real line, (a,b) = {x | a < x < b}.

2.2.7 Lower limit topology \mathbb{R}_{l} on \mathbb{R} : Topology generated by \mathcal{B}’.

2.2.7.1 \mathcal{B}’ is the collection of all half-open intervals

2.2.8 K-topology \mathbb{R}_{K} on \mathbb{R} : Topology generated by \mathcal{B}’’

2.2.8.1 \mathcal{B}’’ is the collection of all open intervals (a,b) , along with all sets of the form (a,b) – K

2.2.8.1.1 K = {x | x = 1/n, for n \in \mathbb{Z}_{+}}

2.2.9 (Lem 13.4) Topologies of \mathbb{R}{l} and \mathbb{R}{K} are strictly finer than the standard topology on \mathbb{R}, but not comparable with one another.

2.2.10 Subbasis \mathcal{S} for a topology on X : a collection of subsets of X whose union equals X

2.2.10.1 Topology generated by the subbasis \mathcal{S} : the collection \Tau of all unions of finite intersections of elements of \mathcal{S}

2.3 Order topology

2.3.1 Order topology : If X is a simply ordered set, there is a standard topology for X, defined using order relation.

2.3.2 Suppose X is a set having a simple order relation <. Given elements a and b of X s.t. a <b,

2.3.2.1 Open interval (a,b) = {x| a<x<b}

2.3.2.2 Closed interval [a,b] = {x | a \le x \le b}

2.3.2.3 Half-open intervals]

2.3.2.3.1 (a,b]= {x | a < x \le b }

2.3.2.3.2 [a,b) = {x | a \le x < b}

2.3.3 Order topology : Let X be a set with simple order relation, assume X has more than one element. Let \mathcal{B} be the collection of all sets of the following types. The coillection \mathcal{B} is a basis for a topology on X, which is order topology.

2.3.3.1 All open intervals in X

2.3.3.2 All intervals of the form [a_{0},b) , where a_{0} is the smallest element (if any) of X.

2.3.3.3 All intervals of the form (a, b_{0}], where b_{0} is the largest element (if any) of X.

2.3.4 If X is an ordered set and a is an element of X, there are four subsets of X that are called rays determined by a.

2.3.4.1 Open rays

2.3.4.1.1 (a, +\infty) = {x | x > a}

2.3.4.1.2 (-\infty, a) = {x | x < a}

2.3.4.2 Closed rays

2.3.4.2.1 [a, + \infty) = {x | x \ge a}

2.3.4.2.2 (-\infty, a] = {x | x \le a}

2.3.4.3 A topology generated using open rays as a subbasis contains the order topology

2.4 Product Topology on X \times Y

2.4.1 Product topology on X \times Y ( topological spaces X and Y) : topology having as basis the collection \mathcal{B} of all sets of the form U \times V, where U is an open subset of X and V is an open subset of Y.

2.4.2 (Thm 15.1) If \mathcal{B} is a basis for the topology of X and \mathcal{C} is a basis for the topology of Y, then the collection \mathcal{D} = {B \times C | B \in \mathcal{B} and C \in \mathcal{C}} is a basis for the topology of X \times Y.

2.4.2.1 Pf) (Lem 13.2)

2.4.3 Projections of X \times Y onto its first and second factors

2.4.3.1 \pi_{1} : X \times Y -> X , \pi_{1} (x,y) = x

2.4.3.2 \pi_{2} : X \times Y -> Y, \pi_{2} (x,y) = y

2.4.4 (Thm 15.2) The collection \mathcal{S} = {\pi^{-1}{1}(U) | U open in X} \cup {\pi^{-1}{2}(V) | V open in Y} is a subbasis for the product topology on X \times Y.

2.5 Subspace topology

2.5.1 Subspace topology : Let X be a topological space with topology \Tau. If Y is a subset of X, the collection \Tau_{Y} = {Y \cap U | U \in \Tau} is a topology on Y.

2.5.1.1 Subspace of X : Y with this topology

2.5.2 (Lem 16.1) If \mathcal{B} is a basis for the topology of X then the collection \mathcal{B}_{Y} = {B \cap Y | B \in \mathcal{B}} is a basis for the subspace topology on Y.

2.5.2.1 Pf) (Lem 13.2)

2.5.3 Open in Y ( a set U ) : U belongs to the topology of Y.

2.5.4 (Lem 16.2) Let Y be a subspace of X. If U is open in Y and Y is open in X, then U is open in X.

2.5.5 (Thm 16.3) If A is a subspace of X and B is a subspace of Y, then the product topology on A \times B is the same as the topology A \times B inherits as a subspace of X \times Y.

2.5.6 Ordered square I^{2}_{0} : set I \times I in the dictionary order topology (I = [0,1])

2.5.6.1 Dictionary order topology of it \neq subset topology inherited from \mathbb{R}^{2}

2.5.7 Convex in X (a subset Y of an ordered set X) : for each pair of points a<b of Y, the entire interval (a,b) of points of X lies in Y.

2.5.8 (Thm 16.4) : Let X be an ordered set in the order topology; Let Y be a subset of X that is convex in X, Then the order topology on Y is the same as the topology Y inherits as a subspace of X.]

2.6 Closed sets and limit points

2.6.1 Closed

2.6.1.1 Closed ( a subset A of a topological space X) : if the set X-A is open

2.6.1.2 (Thm 17.1) Let X be a topological space, then the following conditions hold

2.6.1.2.1 \empty and X are closed.

2.6.1.2.2 Arbitrary intersections of closed sets are closed

2.6.1.2.3 Finite unions of closed sets are closed.

2.6.1.3 Closed in Y ( a set A) : If Y is a subspace of X, if A is a subset of Y and if A is closed in the subspace topology of Y.

2.6.1.4 (Thm 17.2) Let Y be a subspace of X. Then a set A is closed in Y iff it equals the intersection of a closed set of X with Y.

2.6.1.5 (Thm 17.3) Let Y be a subspace of X. If A is closed in Y and Y is closed in X, then A is closed in X.

2.6.2 Closure and Interior of a set

2.6.2.1 Interior ( a subset A of a topological space X) : union of all open sets contained in A

2.6.2.2 Closure ( a subset A of a topological space) : intersection of all closed sets containing A.

2.6.2.3 Int A \subset A \subset \bar{A}

2.6.2.4 (Thm 17.4) Let Y be a subspace of X. Let A be a subset of Y. Let \bar{A} denote the closure of A in X. Then the closure of A in Y equals \bar{A} \cap Y.

2.6.2.4.1 Pf) Thm 17.2

2.6.2.5 Intersects (sets A, B) : the intersection A \cap B is not empty.

2.6.2.6 (Thm 17.5) Let A be a subset of the topological space X.

2.6.2.6.1 x \in \bar{A} iff every open set U containing x intersects A.

2.6.2.6.2 Suppose the topology of X is given by a basis, then x \in \bar{A} iff every basis element B containing x intersects A.

2.6.2.7 U is a neighborhood of x (a set U, an element x) : U is an open set containing x

2.6.3 Limit points

2.6.3.1 Limit point x of A ( a point x in X , a subset A of the topological space X) : every neighborhood of x intersects A in some point other than x itself.

2.6.3.1.1 x is a limit point of A if it belongs to the closure A – {x}

2.6.3.2 (Thm 17.6) Let A be a subset of the topological space X. Let A’ be the set of all limit points of A. Then \bar{A} = A \cup A’.

2.6.3.2.1 Pf) Thm 17.5.

2.6.3.2.2 (Cor) A subset of a topological space is closed iff it contains all its limit points.

2.6.4 Hausdorff spaces

2.6.4.1 Converges to the point x of X ( a sequence x_{1}, x_{2}, … of the points of the space X) : corresponding to each neighborhood U of x, there is a positive integer N s.t. x_{n} \in U for all n \ge N.

2.6.4.2 Hausdorff space ( a topological space X) : for each pair x_{1}, x_{2} of distinct points of X, there exists neighborhoods U_{1}, U_{2} of x_{1} and x_{2}, respectively, that are disjoint.

2.6.4.3 (Thm 17.8) Every finite point set in a Hausdorff space X is closed.

2.6.4.4 T_{1} axiom : Every finite point sets in a space are closed

2.6.4.5 (Thm 17.9) Let X be a space satisfying the T1 axiom. Let A be a subset of X. Then the point x is a limit point of A iff every neighborhood of x contains infinitely many points of A.

2.6.4.6 (Thm 17.10) If X is a Hausdorff space, then a sequence of points of X converges to at most one point of X.

2.6.4.6.1 x_{n} -> x <= Limit x of the sequence x_{n}

2.6.4.7 (Thm 17.11) Every simply ordered set is a Hausdorff space in the order topology. The product of two Hausdorff spaces is a Hausdorff space. A subspace of a Hausdorff space is a Hausdorff space.

2.7 Continuous functions

2.7.1 Continuity of a Function

2.7.1.1 Continuous ( A function f: X ->Y, Topological spaces X , Y) : For each open subset V of Y, the set f^{-1}(V) is an open subset of X.

2.7.1.1.1 F is continuous relative to specific topologies on X and Y.

2.7.1.2 (Thm 18.1) Let X and Y be topological spaces. Let f : X ->Y. then the following are equivalent.

2.7.1.2.1 f is continuous.

2.7.1.2.2 For every subset A of X, one has f(\bar{A}) \subset \bar{f(A)}.

2.7.1.2.3 For every closed set B of Y, the set f^{-1}(B) is closed in X.

2.7.1.2.4 For each x \in X and each neighborhood V of f(x), there is a neighborhood U of x s.t. f(U) \subset V.

2.7.1.2.4.1 f is continuous at the point x.

2.7.2 Homeomorphism

2.7.2.1 Homeomorphism ( f : X -> Y, topological spaces X, Y) : Let f be bijection. Both the function and the inverse function f^{-1} : Y ->X are continuous,

2.7.2.2 Topological property of X : any property of X expressed in terms of the topology of X yields, via the correspondence f, the corresponding property for the space Y.

2.7.2.3 Topological imbedding f of X in Y ( topological spaces X and Y, f : X->Y injective) : the function f’ : X -> f(X) , which is bijective, happens to be homeomorphism

2.7.2.4 Unit circle s^{1} = { x \times y | x ^{2} + y^{2} = 1} , [0,1)

2.7.2.4.1 F : [0,1) -> s^{1} , (cos 2\pi t, sin 2\pi t)

2.7.3 Constructing Continuous functions

2.7.3.1 (Rules for constructing continuous functions)(Thm 18.2) : Let X, Y, Z be topological spaces.

2.7.3.1.1 (Constant function) If f : X->Y maps all of X into the single point y_{0} of Y, then f is continuous.

2.7.3.1.2 (Inclusion) If A is a subspace of X, the inclusion function j : A ->X is continuous.

2.7.3.1.3 (Composites) If f : X->Y and g : Y ->Z are continuous, then the map g \bullet f : X -> Z is continuous.

2.7.3.1.4 (Restricting the domain) If f : X->Y is continuous, and if A is a subspace of X, then the restricted function f | A : A->Y is continuous.

2.7.3.1.5 (Restricting or expanding the range) Let f : X -> Y be continuous. If Z is a subspace of Y containing the image set f(X), then the function g : X ->Z obtained by restricting the range of f is continuous. If Z is a space having Y as a subspace, then the function h: X ->Z obtained by expanding the range of f is continuous.

2.7.3.1.6 (Local formulation of continuity) The map f: X->Y is continuous if X can be written as the union of open sets U_{\alpha} s.t. f | U_{\alpha} is continuous for each \alpha.

2.7.3.2 (The pasting lemma)(Thm 18.3) : Let X = A \cup B, where A and B are closed in X. Let f : A -> Y and g : B -> Y be continuous. If f (x) = g (x) for every x \in A \cap B, then f and g combine to give a continuous function h : X -.Y, defined by setting h(x) = f(x) if x \in A, and h(x) = g(x) if x \in B.

2.7.3.3 (Map into products) (Thm 18.4) Let f : A -> X \times Y be given by the equation f(a) = (f_{1}(a), f_{2}(a)). Then f is continuous iff the function f_{1} : A -> X and f_{2} : A -> Y are continuous.

2.7.3.3.1 Maps f_{1} and f_{2} are called the coordinate functions of f.

2.7.3.4 (Uniform Limit Theorem on a Real space) : If a sequence of continuous real-valued functions of a real variable converges uniformly to a limit function, then the limit function is necessarily continuous.

2.8 Product Topology

2.8.1 J-tuple of elements of X ( an Index set J, a set X) : a function \mathbf{x} : J -> X

2.8.1.1 \alpha th coordinate x_{\alpha} of \mathbf{x} : the value of \mathbf{x} at \alpha

2.8.1.2 (x_{\alpha})_{\alpha \in J}  \mathbf{x}

2.8.2 Cartesian product \prod_{\alpha \in J} A_{\alpha} ( an indexed family {A_{\alpha}}{\alpha \in J} ) : a set of all J-tuples (x{\alpha}){\alpha \in J} of elements of X s.t. x{\alpha} \in A_{\alpha} for each \alpha \in J.

2.8.2.1 Set of all functions \mathbf{x} : J -> \bigcup_{\alpha \in J} A_{\alpha} s.t \mathbf{x} (\alpha) \in A_{\alpha} for each \alpha \in J.

2.8.3 Box topology( {X_{\alpha}}_{\alpha \in J}) : the topology generated by this basis

2.8.3.1 Basis for a topology on the product space \prod_{\alpha \in J} X_{\alpha} the collection of all sets of the form \prod_{\alpha \in J} U_{\alpha} where U_{\alpha} is open in X_{\alpha}.

2.8.3.2 Projection mapping associated with the index \beta : \pi_{\beta}((x_{\alpha}){\alpha \in J}) = x{\beta}

2.8.4 Product topology : The topology generated by the subbasis \mathcal{S}

2.8.4.1 \mathcal{S} = \bigcup_{\beta \in J} \mathcal{S}_{\beta}

2.8.4.1.1 \mathcal{S}{\beta} = {\pi^{-1}{\beta}(U_{\beta}) | U_{\beta} open in X_{beta}}

2.8.4.2 Product space : \prod_{\alpha \in J} X_{\alpha}

2.8.5 (Comparison of the box and product topologies) (Thm 19.1) : The box topology on \prod X_{\alpha} has as basis all sets of the form \prod U_{\alpha}, where U_{\alpha} is open in X_{\alpha} for each \alpha . The product topology on \prod X_{\alpha} has as basis all sets of the form \prod U_{\alpha} , where U_{\alpha} is open in X_{\alpha} for each \alpha and U_{\alpha} equals X_{\alpha} except for finitely many values of \alpha.

2.8.5.1 Box topology is finer than the product topology, but pretty much the same.

2.8.5.2 When considering the product \prod X_{\alpha}, the product topology is usually assumed.

2.8.6 (Thm 19.2) Suppose the topology on each space X_{\alpha} is given by a basis \mathcal{B}{\alpha}. The collection of all sets of the form \prod{\alpha \in J} B_{\alpha} where B_{\alpha} \in \mathcal{B}{\alpha} for each \alpha, will serve as a basis for the box topology on \prod{\alpha \in J} X_{\alpha}.

2.8.6.1 The collection of all sets of the same form, where B_{\alpha} \in \mathcal{B}{\alpha} for finitely many indices \alpha and B{\alpha} = X_{\alpha} for all the remaining indices, will serve as a basis for the product topology \prod_{\alpha \in J} X_{\alpha}.

2.8.7 (Thm 19.3) Let A_{\alpha} be a subspace of X_{\alpha}. For each \alpha \in J. Then \prod A_{\alpha} is a subspace of \prod X_{\alpha} if both products are given the box topology, or if both products are given the product topology.

2.8.8 (Thm 19.4) If each space X_{\alpha}is a Hausdorff space, then \prod X_{\alpha} is a Hausdorff space in both the box and product topologies.

2.8.9 (Thm 19.5) Let {X_{\alpha}} be an indexed family of spaces ; let A_{\alpha} \subset X_{\alpha} for each \alpha. If \prod X_{\alpha} is given either the product or the box topology, then \prod \bar{A_{\alpha}} = \bar{\prod A_{\alpha}}.

2.8.10 (Thm 19.6) Let f: A => \prod_{\alpha \in J} X_{\alpha} be given by e equation f(a) = (f_{\alpha} (a) ) {\alpha \in J} where f{\alpha} : A -> X_{\alpha} for each \alpha . Let \prod X_{\alpha} have the product topology, then the function f is continuous iff each function f_{\alpha} is continuous.

2.9 Metric topology

2.9.1 Metiric on a set X : a function d : X \times X -> R having the following properties.

2.9.1.1 d(x,y) \ge 0 for all x, y \in X; equality iff x = y.

2.9.1.2 d(x,y) = d(y,x) for all x,y \in X.

2.9.1.3 (Triangle inequality) d(x,y) + d(y,z) \ge d(x,z) for all x,y,z \in X.

2.9.2 Distance between x and y : d(x,y)

2.9.2.1 B_{d} (x,\epsilon) : \epsilon -ball centered at x

2.9.3 Metric topology induced by d ( a metric d on the set X) : Collection of all \epsilon -balls B_{d} (x,\epsilon) for x \in X and \epsilon > 0 being a basis for a topology on X.

2.9.4 Metrizable (a topological space X) : there exists a metric d on the set X that induces the topology of X.

2.9.5 Metric space : a metrizable space X together with a specific metric d that gives the topology of X.

2.9.6 Bounded ( a subset A of a metric space X with metric d) : there is some number M s.t. d(a_{1}, a_{2}) \le M for every pair a_{1}, a_{2} of points of A.

2.9.6.1 Diameter of A : if A is bounded and nonempty, diam A = sup{d(a_{1}, a_{2}) | a_{1}, a_{2} \subset A}

2.9.7 Standard bounded metric corresponding to d ( a metric space X with metric d) : \bar{d} : X \times X -> \mathbb{R} by the equation \bar{d} (x,y) = min{d(x,y) , 1}. Then \bar{d} is a metric that induces the same topology as d.

2.9.8 Norm of \mathbf{x} (\mathbf{x} = (x_{1}, …, x_{n} ) in \mathbb{R}^{n} ) : \Vert x \Vert = ( x^{2}{1} + … + x^{2}{n} )^{frac{1}{2}}

2.9.8.1 Euclidean metric d on \mathbb{R}^{n} : d(x,y) = \Vert \mathbf{x} - \mathbf{y} \Vert = [(x_{1} – y_{1})^{2} + … + (x_{n} – y_{n})^{2}]^{frac{1}/{2}}

2.9.8.2 Squate metric \rho : \rho (\mathbf{x}, \mathbf{y}) = max{ \vert x_{1} – y_{1} \vert , …, \vert x_{n} – y_{n} \vert }

2.9.9 (Lem 20.2) Let d and d’ be two metrics on the set X; Let \Tau and \Tau’ be the topologies they induce, respectively. Then \Tau’ is finer than \Tau iff for each x in X and each \epsilon >0, there exists a \delta >0 s.t. B_{d’} (x,\delta) \subset B_{d} (x,\epsilon).

2.9.9.1 Pf) (Lem 13.3)

2.9.10 (Thm 20.3) The topologies on \mathbb{R}^{n} induced by the Euclidean metric d and the squate metric \rho are the same as the product topology on \mathbb{R}^{n}.

2.9.11 Uniform metric on \mathbb{R}^{J} ( an index set J) : Given points \mathbf{x} = (x_{\alpha}){\alpha \in J} and \mathbf{y} = (y{\alpha}){\alpha \in J} of \mathbb{R}^{J}. Define a metric \bar{\rho} on \mathbb{R}^{J} by the equation \bar{\rho} (\mathbf{x} , \mathbf{y}) = sup{\bar{d} (x{\alpha} , y_{\alpha}) | \alpha \in J}, where \bar{d} is the standard bounded metric on \mathbb{R}.

2.9.11.1 Uniform topology : Topology \bar{\rho} induces

2.9.12 (Thm 20.4) Uniforn topology on \mathbb{R}^{J} is finer than the product topology and coarser than the box topology; these three topologies are all different if J is infinite.

2.9.13 (Thm 20.5) Let \bar{d} (a,b) = min{\vert a-b \vert , 1} be the standard bounded metric on \mathbb{R}. if \mathbf{x} and \mathbf{y} are two points of \mathbb{R}^{\omega}, define D(x,y) = sup{frac{\bar{d} (x_{i}, y_{i})} {i}}. Then D is a metric that induces the product topology on \mathbb{R}^{\omega}.

2.10 Metric topology continued

2.10.1 (Thm 21.1) Let f : X->Y; let X and Y be metrizable with metrics d_{X} and d_{Y}, respectively. Then continuity of f is equivalent to the requirement that given x \in X and given \epsilon > 0, there exists \delta > 0 s.t. d_{X} (x,y) < \delta -> d_{Y} (f_{x}, f_{y}) < \epsilon.

2.10.2 (Sequenece Lemma) (Lem 21.2) : Let X be a topological space; let A \subset X. if there is a sequence of points of A converging to x, then x \in \bar{A}; the converse holds if X is metrizable.

2.10.2.1 Pf) Thm 17.5

2.10.3 (Thm 21.3) Let f : X -> Y. If the function f is continuous, then for every convergent sequence x_{n} -> x in X, the sequence f(x_{n}) converges to f(x). the converse holds if X is metrizable.

2.10.3.1 Pf) Sequence lemma

2.10.4 a countable basis at the point x ( a space X) : there is a countable collection {U_{n}}{n \in \mathbb{Z}{+}} of neighborhoods of x s.t. any neighborhood U of x contains at least one of the sets U_{n}.

2.10.5 First countability axiom : A space X that has a countable basis at each of its points

2.10.6 (Lem 21.4) Addition, subtraction, and multiplication operations are continuous functions from \mathbb{R} \times \mathbb{R} into \mathbb{R}, and the quotient operation is a continuous function from \mathbb{R} \times (\mathbb{R} – {0}) into \mathbb{R}.

2.10.7 (Thm 21.5) If X is a topological space, and if f,g : X -> \mathbb{R} are continuous functions, then f + g, f – g, and f \cdot g are continuous. If g(x) \neq 0 for all x, then f/g is continuous.

2.10.7.1 Pf) (Thm 18.4)

2.10.8 Converges uniformly to the function f: X -> Y ( f_{n} : X -> Y a sequence of functions from the set X to the metric space Y) : Let d be the metric for Y. Given \epsilon >0, there exists an integer N s.t. d(f_{n} (x) , f (x)) < \epsilon for all n > N and all x in X;

2.10.9 (Uniform limit theorem) (Thm 21.6) : Let f_{n} : X -> Y be a sequence of continuous functions from the topological space X to the metric space Y. if (f_{n}) converges uniformly to f, then f is continuous.

2.10.10 Examples of spaces not metrizable

2.10.10.1 \mathbb{R}^{\omega} in the box topology is not metrizable.

2.10.10.2 \An uncountable product of \mathbb{R} with itself is not metrizable.

2.11 Quotient topology

2.11.1 Quotient map ( The map p) : Let X and Y be topological spaces; let p : X -> Y be a surjective map. If, A subset U of Y is open in Y iff p^{-1}(U) is open in Y, p is a quotient map.

2.11.2 Saturated with respect to the surjective map p : X -> Y ( a subset C of X) : C contains every set p^{-1}({y}) that it intersects.

2.11.3 If p : X->Y is a surjective continuous map that is either open or closed, then p is a quotient map.

2.11.3.1 Open map (a map f : X -> Y) : for each open set U of X the set f(U) is open in Y

2.11.3.2 Closed map (a map f : X -> Y) : for each closed set A of X the set f(A) is closed in Y

2.11.4 Quotient topology induced by p : If X is a space and A is a set and if p : X -> A is a surjective map, then there exists exactly one topology \Tau on A relative to which p is a quotient map.

2.11.5 Quotient space of X (a topological space X) : Let X* be a partition of X into disjoint subsets whose union is X. Let p : X -> X* be the surjective map that carries each point of X to the element of X* containing it. In the quotient topology induced by p, the space X* is called a quotient space of X.

2.11.6 (Thm 22.1) Let p : X->Y be a quotient map; Let A be a subspace of X that is saturated with respect to p; let q : A -> p(A) be the map obtained by restricting p.

2.11.6.1 If A is either open or closed in X, then q is a quotient map.

2.11.6.2 If p is either an open map or a closed map, then q is a quotient map.

2.11.7 (Thm 22.2) Let p : X -> Y be a quotient map. Let Z be a space and let g : X -> Z be a map that is constant on each set \rho^{-1}({y}), for y \in Y. Then g induces a map f : Y -> Z s.t. f \bullet p = g. The induced map f is continuous iff g is continuous; f is a quotient map iff g is a quotient map.

2.11.7.1 (Cor) Let g : X ->Z be a surjective continuous map. Let X* be the following collection of subsets of X : X* = {g^{-1}({z}) | z \in Z}. Give X* the quotient topology.

2.11.7.1.1 The map g induces a bijective continuous map f: X* -> Z, which is a homeomorphism iff g is a quotient map.

2.11.7.1.2 If Z is Hausdorff, so is X*.

Connectedness and Compactness

3 Connectedness and Compactness

3.1 Connected spaces :

3.1.1 Separation of X ( a topological space X) : a pair U, V of disjoint nonempty open subsets of X whose union is X.

3.1.2 Connected ( a topological space X) if there does not exist a separation of X.

3.1.2.1 The only subsets of X that are both open and closed in X are the empty set and X itself.

3.1.3 (Lem 23.1) If Y is a subspace of X, a separation of Y is a pair of disjoint nonempty sets A and B whose union is Y, neither of which contains a limit point of the other. The space Y is connected if there exists no separation of Y.

3.1.4 (Lem 23.2) If the sets C and D form a separation of X, and if Y is a connected subspace of X then Y lies entirely within either C or D.

3.1.5 (Thm 23.3) The union of a collection of connected subspaces of X that have a point in common is connected.

3.1.6 (Thm 23.4) Let A be a connected subspace of X. If A \subset B \subset \bar{A}, then B is also connected.

3.1.6.1 Pf) Lem 23.2

3.1.7 (Thm 23.5) The image of a connected space under a continuous map is connected.

3.1.8 (Thm 23.6) A finite cartesian product of connected spaces is connected.

3.1.9 An arbitrary product of connected spaces is connected in the product topology.

3.2 Connected subspaces of the real line.

3.2.1 Linear continnum ( a simply ordered set L) : if following holds.

3.2.1.1 L has the least upper bound property.

3.2.1.2 If x < y, there exists z s.t. x < z < y.

3.2.2 (Thm 24.1) If L is a linear continuum in the order topology, then L is connected, and so are intervals and rays in L.

3.2.2.1 Pf) convex

3.2.2.2 (Cor) The real line \mathbb{R} is connected and so are intervals and rays in \mathbb{R}.

3.2.3 (Intermediate value theorem ) (Thm 24.3): Let f : X -> Y be a continuous map, where X is a connected space and Y is an ordered set in the oerder topology. If a and b are two points of X and if r is a point of Y lying between f(a) and f(b), then there exists a point c of X s.t. f(c) = r

3.2.4 Path in X from x to Y ( points x, y of the space X) : continuous map f : [a,b] -> X of some closed interval in the real line into X, s.t. f(a) = x and f(b) = y.

3.2.5 Path connected (a space X) : every pair of points of X can be joined by a path in X.

3.2.5.1 Path connected space X is connected.

3.2.6 Examples

3.2.6.1 Unit ball B^{n} in \mathbb{R}^{n} : B^{n} = {\mathbf{x} | \Vert x \Vert \le 1}

3.2.6.1.1 \Vert \mathbf{x} \Vert = \Vert (x_{1}, …, x_{n}) \Vert = (x^{2}{1} + .. + x^{2}{n} )^{fact{1}{2}}

3.2.6.1.2 Unit ball is path connected.

3.2.6.2 Punctured Euclidean space : \mathbb{R}^{n} – {\mathbf{0}}

3.2.6.2.1 \mathbf{0} : origin of \mathbb{R}^{n}.

3.2.6.2.2 If n > 1, it is path connected.

3.2.6.3 Unit sphere S^{n-1} in \mathbb{R}^{n} : S^{n-1} = {\mathbf{x} | \Vert \mathbf{x} \Vert = 1}

3.2.6.3.1 If n > 1, it is path connected

3.2.6.4 Ordered square I^{2}_{0} is connected but not path connected.

3.2.6.5 Topologist’s sine curve : S = {x \times sin(1/x) | 0<x \le 1}

3.2.6.5.1 Connected but not path connected

3.3 Components and Local connectedness

3.3.1 components of X (a space X) : Equivalent classes where an equivalence relation on X by setting x ~ y if there is a connected subspace of X containing both x and y.

3.3.1.1 Also called connected components

3.3.2 (Thm 25.1) The components of X are connected disjoint subspaces of X whose union is X, s.t. each nonempty connected subspace of X intersects only one of them.

3.3.3 Path components of X (a space X) : Equivalent classes where an equivalence relation on X by defining x ~ y if there is a path in X from x to y.

3.3.3.1 Pf ) (pasting lemma)

3.3.4 (Thm 25.2) The path components of X are path-connected disjoint subspaces of X whose union is X, s.t. each nonempty path connected subspace of X intersects only one of them.

3.3.5 Locally connected at x (a point x in a space X) : For every neighborhood U of x, there is a connected neighborhood V of x contained by U.

3.3.6 Locally connected (a space X): X is locally connected at each of its points.

3.3.7 Locally path connected at x (a point x in a space X) : For every neighborhood U of x, there is a path connected neighborhood V of x contained by U.

3.3.8 Locally path connected (a space X): X is locally path connected at each of its points.

3.3.9 (Thm 25.3) A space X is locally connected iff for every open set U of X, each component of U is open in X.

3.3.10 (Thm 25.4) A space X is locally path connected iff for every open set U of X, each path component of U is open in X.

3.3.11 (Thm 25.5) If X is a topological space, each path component of X lies in a component of X. If X is a locally path connected, then the components and the path components of X are the same.

3.4 Compact spaces

3.4.1 Covering of X (a collection \mathcal{A} of subsets of a space X) : the union of the elements of \mathcal{A} is equal to X.

3.4.1.1 Also called ‘to cover X’.

3.4.1.2 Open covering of X if its elements are open subsets of X.

3.4.2 Compact (a space X) : every open covering \mathcal{A} of X contains a finite subcollection that also covers X

3.4.3 Cover Y (a subspace Y of X) : If, for a collection \mathcal{A} of subsets of X, the union of its elements contains Y.

3.4.4 (Lem 26.1) Let Y be a subspace of X. Then Y is compact iff every covering of Y by sets open in X contains a finite subcollection covering Y.

3.4.5 (Thm 26.2) Every closed subspace of a compact space is compact.

3.4.6 (Thm 26.3) Every compact subspace of a Hausdorff space is closed.

3.4.7 (Lem 26.4) If Y is a compact subspace of the Hausdorff space X and x_{0} is not in Y, then there exist disjoint open sets U and V of X containing x_{0} and Y, tespectively.

3.4.8 (Thm 26.5) The image of a compact space under a continuous map is compact.

3.4.9 (Thm 26.6) Let f : X -> Y be a bijective continuous function. If X is compact and Y is Hausdorff, then f is a homeomorphism.

3.4.9.1 (Thm 26.2) (Thm 26.5) (Thm 26.3)

3.4.10 (Thm 26.7) The product of finitely many compact spaces is compact)

3.4.10.1 (Tube Lemma) (Lem 26.8) : Consider the product space X \times Y, where Y is compact. If N is an open set of X \times Y containing the slice x_{0} \times Y of X \times Y, then N contains some tube W \times Y about x_{0} \times Y, where W is a neighborhood of x_{0} in X.

3.4.11 Finite intersection property (a collection \mathcal{C} of subsets of X) : For every finite subcollection {C_{1}, …, C_{n}} of \mathcal{C}, the intersection C_{1} \cap … \cap C_{n} is nonempty.

3.4.12 (Thm 26.9) Let X be a topological space. Then X is compact iff for every collection \mathcal{C} of closed sets in X having the finite intersection property, the intersection \bigcup_{C \in \mathcal{C}} C of all the elements of \mathcal{C} is nonempty.

3.4.12.1 Nested sequence C_{1} \supset C_{2} \supset … \supset C_{n} \supset .. of closed sets in a compact space X

3.5 Compact subspaces of the Real line

3.5.1 (Thm 27.1) Let X be a simply ordered set having the least upper bound property. In the order topology, each closed interval in X is compact.

3.5.1.1 Pf) Subspace topology equals order topology

3.5.1.2 (Cor) Every closed interval in \mathbb{R} is compact.

3.5.2 (Thm 27.3) A subspace A of \mathbb{R}^{n} is compact iff it is closed and is bounded in the Euclidean metric d or the square metric \rho.

3.5.3 (Extreme value theorem) (Thm 27.4) Let f: X -> Y be continuous, where Y is an ordered set in the order topology. If X is compact, then there exists points c and d in X s.t. f(c) \le f(x) \le f(d) for every x \in X.

3.5.4 Distance from x to A ( a point x \in X, nonempty subset A of X) : For a metric space (X,d), d(x,A) = inf {d(x,a) | a \in A}

3.5.4.1 D(x,A) is continuous

3.5.5 (Lebesgue number lemma)(Lem 27.5) Let \mathcal{A} be an open covering of the metric space (X,d). If X is compact, there is a \delta > 0 s.t. for each subset of X having diameter less than \delta, there exists an element of \mathcal{A} containing it.

3.5.5.1 Lebesgue number for the covering \mathcal{A} : \delta

3.5.6 Uniformly continuous (a function f : X -> Y) : Let metric spaces (X,d_{X}), (Y,d_{y}), if given \epsilon >0, there is a \delta >0 s.t. for every pair of points x_{0}, x_{1} of X, d_{X} (x_{0}, x_{1}) < \delta -> d_{Y} (f (x_{0}), f (x_{1}) ) < \epsilon

3.5.7 (Uniform continuity theorem) (Thm 27.6) : Let f : X -> Y be a continuous map of the compact metric space (X,d_{X}) to the metric space (Y,d_{Y}). Then f is uniformly continuous.

3.5.7.1 Pf) (Lebesgue number lemma)

3.5.8 Isolated point x of X (a point x of a space X) : one-point set {x} is open in X.

3.5.9 (Thm 27.7) Let X be a nonempty compact Hausdorff space. If X has no isolated points, then X is uncountable.

3.5.9.1 Pf) (Thm 26.9)

3.5.9.2 (Cor 27.8) Every closed interval in \mathbb{R} is uncountable.

3.6 Limit point compactness

3.6.1 Limit point compact (a space X) : every infinite subset of X has a limit point.

3.6.2 (Thm 28.1) Compactness implies limit point compactness, but not conversely.

3.6.3 Subsequence ( a sequence (x_{n}) of points of a topological space X) : n_{1} < n_{2} < … < n_{i} < … is an increasing sequence of positive integer, then the sequence (yi) defined by setting y_{i} = x_{n}

3.6.4 Sequentially compact ( a space X) : every sequence of points of X has a convergent subsequence

3.6.5 (Thm 28.2) Let X be a metrizable space. Then the following are equivalent.

3.6.5.1 X is compact.

3.6.5.2 X is limit point compact.

3.6.5.3 X is sequentially compact.

3.6.5.4 Pf) If X is sequentially compact, the (Lebesgue number lemma) holds for X.

3.7 Local compactness

3.7.1 Locally compact at x ( a point x in a space X) : there is some compact subspace C of X that contains a neighborhood of x.

3.7.2 Locally compact (a space X) : X is locally compact at each of its points

3.7.3 (Thm 29.1) Let X be a space. Then X is locally compact Hausdorff iff there exists a space Y satisfying the following conditions. If Y and Y’ are two spaces satisfying these conditions, then there is a homeomorphism of Y with Y’ that equals the identity map on X.

3.7.3.1 X is a subspace of Y.

3.7.3.2 The set Y-X consists of a single point.

3.7.3.3 Y is a compact Hausdorff space.

3.7.4 Compactification Y of X ( a compact Hausdorff space Y and a proper subspace X of Y) : closure of X equals Y

3.7.5 One-point compactification Y of X( a compact Hausdorff space Y and a proper subspace X of Y ): Y-X equals a single point

3.7.5.1 X has a one-point compactification Y iff X is a locally compact Hausdorff space that is not itself compact.

3.7.5.2 Riemann sphere : \mathbb{C} \cup {\infty} , one-point compactification of \mathbb{R}^{2}

3.7.6 (Thm 29.2) Let X be a Hausdorff space. Then X is locally compact iff given x in X, and given a neighborhood U of x, there is a neighborhood V of x s.t. \bar{V} is compact and \bar{V} \subset U.

3.7.6.1 (Cor 29.3) Let X be locally compact Hausdorff; let A be a subspace of X. If A is closed in X or open in X, then A is locally compact.

3.7.6.2 (Cor 29.4) A space X is homeomorphic to an open subspace of a compact Hausdorff space iff X is locally compact Hausdorff.

3.7.6.2.1 Pf) (Thm 29.1) (Cor 29.3)

Countability and Separation Axioms

4 Countability and Separation Axioms

4.1 Countability axioms

4.1.1 a countable basis at the point x ( a space X) : there is a countable collection \mathcal{B} of neighborhoods of x s.t. any neighborhood U of x contains at least one of the elements of \mathcal{B}.

4.1.2 First countability axiom : A space X that has a countable basis at each of its points

4.1.2.1 Also called ‘first-countable’

4.1.3 (Thm 30.1) Let X be a topological space.

4.1.3.1 Let A be a subset of X. If there is a sequence of points of A converging to x, then x \in \bar{A} ; the converse holds if X is first-countable.

4.1.3.2 Let f : X -> Y. If f is continuous, then for every convergent sequence x_{n} -> x in X, the sequence f (x_{n}) converges to f(x). The converse holds if X is first-countable.

4.1.4 Second countability axiom : A space X has countable basis for its topology

4.1.4.1 Also called ‘second-countable’

4.1.5 (Thm 30.2) A subspace of a first-countable space is first-countable, and a countable product of first-countable spaces is first-countable. A subspace of a second-countable space is second-countable, and a countable product of second-countable spaces is second-countable.

4.1.6 Dense in X ( a subset A of a space X) : \bar{A} – X

4.1.7 (Thm 30.3) Suppose that X has a countable basis. Then

4.1.7.1 Every open covering of X contains a countable subcollection covering X.

4.1.7.2 There exists a countable subset of X that is dense in X.

4.1.8 Lindelöf space : A space for which every open coveing contains a countable subcovering

4.1.9 Separable (a space) : a space has a countable dense subset

4.1.9.1 Not a good term

4.2 Separation Axioms

4.2.1 Regular ( a space X) : Suppose that one-point sets are closed in X. For each pair consisting of a point x and a closed set B disjoint from x, there exist disjoint open sets containing x and B.

4.2.1.1 Regular space is Hausdorff space.

4.2.2 Normal ( a space x) : Suppose that one-point sets are closed in X. For each pair A, B of disjoint closed sets of X, there exists disjoint open sets containing A and B, respectively.

4.2.2.1 Normal space is Regular space.

4.2.3 (Lem 31.1) Let X be a topological space. Let one-point sets in X be closed.

4.2.3.1 X is regular iff given a point given a point x of X and a neighborhood U of x, there is a neighborhood V of x s.t. \bar{V} \subset U.

4.2.3.2 X is normal iff given al closed set A and an open set U containing A, there is an open set V containing A s.t. \bar{V} \subset U.

4.2.4 (Thm 31.2)

4.2.4.1 A subspace of a Hausdorff space is Hausdorff; a product of Hausdorff spaces is Hausdorff.

4.2.4.2 A subspace of a regular space is regular, a product of regular spaces is regular.

4.2.4.3 Pf) (Lem 31.1), (Thm 19.5)

4.2.5 Example

4.2.5.1 Sorgenfrey plane \mathbb{R}^{2}_{l} is not normal.

4.3 Normal spaces

4.3.1 (Thm 32.1) Every regular space with a countable basis is normal.

4.3.2 (Thm 32.2) Every metrizable space is normal.

4.3.3 (Thm 32.3) Every compact Hausdorff space is normal.

4.3.3.1 Pf) (Lem 26.4)

4.3.4 (Thm 32.4) Every well-ordered set X is normal in the order topology.

4.3.5 Examples not normal

4.3.5.1 If J is uncountable, the product space \mathbb{R}^{J} is not normal.

4.3.5.2 The product space S_{\Omega} \times \bar{S_{\Omega}} is not normal.

4.4 Urysohn lemma

4.4.1 (Urysohn lemma) (Thm 33.1) Let X be a normal space; let A and B be disjoint closed subsets of X. Let [a,b] be a closed interval in the real line. Then there exists a continuous map f : X -> [a,b] s.t. f(x) = a for every x in A, and f(x) = b for every x in B.

4.4.1.1 Pf) (Thm 10.1)

4.4.2 A and B can be separated by a continuous function ( two subsets A,B of the topological space X) : there is a continuous function f : X -> [0,1] s.t. f(A) = {0} and f(B) = {1}.

4.4.3 Completely regular ( a space X) : one-point sets are closed in X and if for each point x_{0} and each closed set A not containing x_{0}, there is a continuous function f : X->[0,1] s.t. f(x_{0}) = 1 and f(A) = {0}

4.4.4 (Thm 33.2) A subspace of a completely regular space is completely regular. A product of completely regular spaces is completely regular.

4.4.4.1 Examples

4.4.4.1.1 \mathbb{R}^{2}{l}, S{\Omega} \times \bar{S_{\Omega}} is completely regular.

4.5 Urysohn Metrization Theorem

4.5.1 (Urysohn Metrization Theorem) (Thm 34.1) Every regular space X with a countable basis is metrizable.

4.5.1.1 Pf) (Thm 20.5) , (There exists a countable collection of continuous functions f_{n} : X -> [0,1] having the property that given any point x_{0} of X and any neighborhood U of x_{0}, there exists an index n s.t. f_{n} is positive at x_{0} and vanishes outside U) , (Urysohn lemma), imbedding

4.5.2 (Imbedding Theorem) (Thm 34.2)

4.5.2.1 Let X be a space in which one-point sets are closed. Suppose that {f_{\alpha}}{\alpha \in J} is an indexed family of continuous function f{\alpha} : X -> \mathbb{R} satisfying the requirement that for each point x_{0} of X and each neighborhood U of x_{0}, there is an index \alpha s.t. f_{\alpha} is positive at x_{0} and vanishes outside U.

4.5.2.1.1 Separate points from closed sets in X : a family of continuous functions behaving as {f_{\alpha}}_{\alpha \in J}

4.5.2.2 Then the function F : X->R^{J} defined by F(x) = (f_{\alpha} (x) )_{\alpha \in J} is an imbedding of X in \mathbb{R}^{J}.

4.5.2.3 If f_{\alpha} maps X into [0,1] for each \alpha, then F imbeds X in [0,1]^{J}.

4.5.3 (Thm 34.3) A space X is completely regular iff it is homeomorphic to a subspace of [0,1]^{J} for some J.

4.6 Tietze Extension Theorem

4.6.1 (Tietze Extension Theorem) (Thm 35.1) : Let X be a normal space; Let A be a closed subspace of X.

4.6.1.1 Any continuous map of A into the closed interval [a,b] of \mathbb{R} may be extended to a continuous map of all of X into [a,b]

4.6.1.2 Any continuous map of A into \mathbb{R} may be extended to a continuous map of all of X into \mathbb{R}.

4.6.1.3 Pf) (Urysohn lemma), (Weierstrass M-test), uniform convergence

4.7 Imbeddings of Manifolds

4.7.1 M-manifold : a Hausdorff space X with a countable basis s.t. each point x of X has a neighborhood that is homeomorphic with an open subset of \mathbb{R}^{m}.

4.7.1.1 Curve : 1-manifold

4.7.1.2 Surface : 2-manifold

4.7.2 Support of \phi (\phi : X -> \mathbb{R}) : closure of the set \phi^{-1} ( \mathbb{R} – {0} )

4.7.3 Partition of unity dominated by {U_{i}} ( a finite index open covering {U_{1}, …, U_{n}} of the space X) : Indexed family of continuous functions \phi_{i} : X -> [0,1] for i = 1, …, n if :

4.7.3.1 (support \phi_{i}) \subset U_{i} for each i.

4.7.3.2 \sum_{i = 1}^{n} \phi_{i} (x) = 1 for each x.

4.7.4 (Existence of finite partitions of unity) (Thm 36.1) : Let {U_{1}, …, U_{n}} be a finite open covering of the normal space X. Then there exists a partition of unity dominated by {U_{i}}.

4.7.4.1 Pf) (Urysohn lemma)

4.7.5 (Thm 36.2) If X is a compact m-manifold, then X can be imbedded in \mathbb{R}^{N} for some positive integer N.

4.7.5.1 Pf) (compact & Hausdorff -> normal)

Tychonoff Theorem

5 Tychonoff Theorem

5.1 Tychonoff Theorem

5.1.1 (Lem 37.1) Let X be a set; let \mathcal{A} be a collection of subsets of X having the finite intersection property. Then there is a collection \mathcal{D} of subsets of X s.t. \mathcal{D} contains \mathcal{A}, and \mathcal{D} has the finite intersection property, and no collection of subsets of X that properly contains \mathcal{D} has this property.

5.1.1.1 A collection \mathcal{D} satisfying the conclusion of this theorem is maximal with respect to the finite intersection property.

5.1.1.2 Pf) (Zorn’s lemma)

5.1.2 (Lem 37.2) Let X be a set; Let \mathcal{D} be a collection of subsets of X that is maximal with respect to the finite intersection property. Then

5.1.2.1 Any finite intersection of elements of \mathcal{D} is an element of \mathcal{D}.

5.1.2.2 If A is a subset of X that intersects every element of \mathcal{D}, then A is an element of \mathcal{D}.

5.1.3 (Tychonoff Theorem) (Thm 37.3) An arbitrary product of compact spaces is compact in the product topology.

5.1.3.1 Pf) (Lem 37.1) (Lem 37.2)

5.2 Stone- \check{C} ech Compactification

5.2.1 Compactification Y of space X : a compact Hausdorff space Y containing X as a subspace s.t. \bar{X} = Y.

5.2.2 Equivalent (Two compactifications Y_{1} and Y_{2} of a space X) : there is a homeomorphism h : Y_{1} -> Y_{2} s.t. h(x) = x for every x \in X.

5.2.3 (Lem 38.1) Let X be a space; suppose that h: X->Z is an imbedding of X in the compact Hausdorff space Z. Then there exists a corresponding compactification Y of X; it has the property that there is an imbedding H : Y -> Z that equals h on X. The compactification Y is uniquely determined up to equivalence.

5.2.3.1 Compactification Induced by imbedding h : Y

5.2.4 (Thm 38.2) Let X be a completely regular space. There exists a compactification Y of X having the property that every bounded continuous map f : X -> \mathbb{R} extends uniquely to a continuous map of Y into \mathbb{R}.

5.2.4.1 Pf) (Tychonoff Theorem), (Thm 34.2) (Lem 38.3)

5.2.5 (Lem 38.3) Let A \subset X; let f : A -> Z be a continuous map of A into the Hausdorff space Z. There is at most one extension of f to a continuous function g:\bar{A} ->Z.

5.2.6 (Thm 38.4) Let X be a completely regular space; let Y be a compactification of X satisfying the extension property of (Thm 38.2). Given any continuous map f: X->C of X into a compact Hausdorff space C, the map f extends uniquely to a continuous map g : Y -> C.

5.2.6.1 Pf) (Completely regular -> imbedded in [0,1]^{J} for some J)

5.2.7 (Thm 38.5) Let X be a completely regular space. If Y_{1} and Y_{2} are two compactifications of X satisfying the extension property of (Thm 38.2), then Y_{1} and Y_{2} are equivalent.

5.2.7.1 Pf) (Preceding theorem)

5.2.8 Stone- \check{C} ech compactification of X ( a completely regular space X) : \beta (X), a compactification of X satisfying the extension condition of (Thm 38.2)

5.2.8.1 Any continuous map f : X->C of X into a compact Hausdorff space C extends uniquely to a continuous map g : \beta (X) -> C.

Metrization Theorems and Paracompactness

6 Metrization Theorems and Paracompactness

6.1 Local finiteness

6.1.1 Locally finite in X (a collection \mathcal{A} of subsets of a topological space X) : Every point of X has a neighborhood that intersects only finitely many elements of \mathcal{A}.

6.1.2 (Lem 39.1) Let \mathcal{A} be a locally finite collection of subsets of X. Then;

6.1.2.1 Any subcollection of \mathcal{A} is locally finite.

6.1.2.2 The collection \mathcal{B} = {\bar{A} }_{A \in \mathcal{A}} of the closures of the elements of \mathcal{A} is locally finite.

6.1.2.3 \bar{\bigcup_{A \in \mathcal{A}}} = \bigcup_{A \in \mathcal{A}} \bar{A}

6.1.3 Locally finite indexed family in X ( an indexed family {A_{\alpha}}{\alpha \in J} ) : every x \in X has a neighborhood that intersects A{\alpha} for only finitely many values of \alpha.

6.1.4 Countably locally finite ( a collection \mathcal{B} of subsets of X) : \mathcal{B} can be written as the countable union of collections \mathcal{B}_{n} , each of which is locally finite.

6.1.5 Refinement \mathcal{B} of \mathcal{A} ( a collection \mathcal{B} of subsets of X) : for each element B of \mathcal{B}, there is an element A of \mathcal{A} containing B.

6.1.5.1 Also called ‘ to refine \mathcal{A}’

6.1.5.2 Open refinement of \mathcal{A} ( a collection \mathcal{B} of subsets of X) : elements of \mathcal{B} are open sets

6.1.5.3 Closed refinement of \mathcal{A} ( a collection \mathcal{B} of subsets of X) : elements of \mathcal{B} are closed sets

6.1.6 (Lem 39.2) Let X be a metrizable space. If \mathcal{A} is an open covering of X, then there is an open covering \Epsilon of X that is countably locally finite.

6.1.6.1 Pf) (well-ordering theorem)

6.2 Nagata-Smirnov Metrization Theorem

6.2.1 G_{\delta} set in X ( a subset A of a space X) : it equals the intersection of a countable collection of open subsets of X.

6.2.2 (Lem 40.1) Let X be a regular space with a basis \mathcal{B} that is countably locally finite. Then X is normal, and every closed set in X is a G_{\delta} set in X.

6.2.2.1 Pf) (Lem 39.1) (Thm 32.1)

6.2.3 (Lem 40.2) Let X be normal; let A be a closed G_{\delta} set in X. Then there is a continuous function f : X-> [0,1] s.t. f(x) = 0 for x \in A and f(x) > 0 for x \notin A.\

6.2.4 (Nagata-Smirnov Metrization Theorem) (Thm 40.3) : A space X is metrizable iff X is regular and has a basis that is countably locally finite.

6.2.4.1 Pf) (Lem 39.2)

6.3 Paracompactness

6.3.1 Paracompact ( a space X) : every open covering \mathcal{A} of X has a locally finite open refinement \mathcal{B} that covers X.

6.3.1.1 \mathbb{R}^{n}

6.3.2 (Thm 41.1) Every paracompact Hausdorff space X is normal.

6.3.3 (Thm 41.2) Every closed subspace of a paracompact space is paracompact.

6.3.4 (Lem 41.3) Let X be regular. Then the following conditions on X are equivalent; Every open covering of X has a refinement that is;

6.3.4.1 An open covering of X and countably locally finite.

6.3.4.2 A covering of X and locally finite.

6.3.4.3 A closed covering of X and locally finite

6.3.4.4 An open covering of X and locally finite

6.3.4.5 Pf) (Lem 39.1)

6.3.5 (Thm 41.4) Every metrizable space is paracompact

6.3.5.1 Pf) (Lem 39.2)

6.3.6 (Thm 41.5) Every regular Lindelöf space is paracompact.

6.3.6.1 Pf) (Preceding lemma)

6.3.7 Partition of unity on X, dominated by {U_{\alpha}} ( a indexed open covering {U_{\alpha}}{\alpha \in J} of the space X) : Indexed family of continuous functions \phi{\alpha} : X -> [0,1] for i = 1, …, n if :

6.3.7.1 (support \phi_{\alpha}) \subset U_{\alpha} for each \alpha

6.3.7.2 The indexed family {Support \phi_{\alpha} } is locally finite

6.3.7.3 \sum \phi_{\alpha} (x) = 1 for each x.

6.3.8 (Shrinking lemma)(Lem 41.6) Let X be a paracompact Hausdorff space; Let {U_{\alpha}}{\alpha \in J} be an indexed family of open sets covering X. Then there exists a locally finite indexed family {V{\alpha}}{\alpha \in J} of open sets covering X s.t. \bar{V}{\alpha} \subset U_{\alpha} for each \alpha.

6.3.8.1 Family {\bar{V}{\alpha}} is a ‘precise refinement’ of the faminy {U{\alpha}} \bar{V}{\alpha} \subset U{\alpha} for each \alpha.

6.3.9 (Thm 41.7) Let X be a paracompact Hausdorff space; let {U_{\alpha}}{\alpha \in J} be an indexed open covering of X. Then there exists a partition of unity on X dominated by {U{\alpha}}.

6.3.9.1 Pf) (Shrinking lemma)

6.3.10 (Thm 41.8) Let X be a paracompact Hausdorff space; let \mathcal{C} be a collection of subsets of X; for each C \in \mathcal{C}, let \epsilon_{C} be a positive number. If \mathcal{C} is locally cinite, there is a continuous function f: X->\mathbb{R} s.t. f(x) >0 for all x, and f(x) \le \epsilon_{C} for x \in C.

6.4 Smirnov Metrization Theorem

6.4.1 Locally metrizable (a space X) : Every point x of X has a neighborhood U that is metrizable in the subspace topology.

6.4.2 (Smirnov metrization theorem) (Theorem 42.1) Space X is metizable iff it is a paracompact Hausdorff space that is locally metrizable.

6.4.2.1 Pf) similar to (Thm 40.3)

Complete Metric spaces and Function spaces

7 Complete Metric spaces and Function spaces

7.1 Complete Metric Spaces

7.1.1 Cauchy sequence (x_{n}) in (X,d) ( a sequence (x_{n}) of points of X, a metric space ( X,d)) : it has the property that given \epsilon > 0 , there is an integer N s.t. d(x_{n}, x_{m}) < \epsilon whenever n, m \ge N

7.1.2 Complete ( a metric space (X,d) ) : Every Cauchy sequence in X converges.

7.1.2.1 If X is complete under the metric d, then X is complete under the standard bounded metric corresponding to d.

7.1.3 (Lem 43.1) : A metric space X is complete if every Cauchy sequence in X has a convergent subsequence.

7.1.4 (Thm 43.2) Euclidean space \mathbb{R}^{k} is complete in either of its usual metrics, the Euclidean metric d or the square metric \rho.

7.1.4.1 Pf) (Thm 28.2)

7.1.5 (Lem 43.3) Let X be the product space X = \prod X_{\alpha} ; let x_{n} be a sequence of points of X. Then x_{n} -> X iff \pi_{\alpha} (\mathbf{x}{n}) -> \pi{\alpha} (\mathbf{x}) for each \alpha.

7.1.6 (Thm 43.4) There is a metric for the product space \mathbb{R}^{\omega} relative to which \mathbb{R}^{\omega} is complete.

7.1.6.1 Pf) D(x,y) = sup{\bar{d}(x_{i}, y_{i}) / i}

7.1.7 Uniform metric on Y^{J} corresponding to the metric d on Y : \bar{\rho} (\mathbf{x}, \mathbf{y}) = sup{\bar{d} (x_{\alpha}, y_{\alpha}) | \alpha \in J)

7.1.7.1 Let (Y,d) be a metric space; let \bar{d} (a,b) = min{d(a,b), 1} be the standard bounded metric on Y derived from d. If \mathbf{x} = (x_{\alpha}){a \in J} and \mathbf{y} = (y{\alpha})_{a \in J} are points of the cartesian product Y^{J}.

7.1.8 (Thm 43.5) If the space Y is complete in the metric d, then the space Y^{J} is complete in the uniform metric \bar{\rho} corresponding to d.

7.1.8.1 Pf) (if (Y,d) is complete, so is (Y, \bar{d}))

7.1.9 Bounded ( a function f : X -> Y) : its image f(X) is a bounded subset of the metric space (Y,d)

7.1.10 (Thm 43.6) Let X be a topological space and let (Y,d) be a metric space. The set \mathcal{C} (X,Y) of continuous functions is closed in Y^{X} under the uniform metric. So is the set \mathcal{B} (X,Y) of bounded functions. Therefore, if Y is complete, these spaces are complete in the uniform metric.

7.1.10.1 Pf) (Uniform limit theorem, Thm 21.6)

7.1.11 Sup metric ( a metric \rho) : \rho ( f, g) = sup { d(f(x), g(x)) | x \in X} If (Y,d) is a metric space, one can define another metric on the set \mathcal{B} (X,Y) of bounded functions from X to Y . \rho is well-defined, for the set f(X) U g(X) is bounded if both f(x) and g(X) are.

7.1.11.1 If X is a compact space, then every continuous function f: X -> Y is bounded, hence the sup metric is defined on \mathcal{C} (X, Y)

7.1.12 (Thm 43.7) Let (X,d) be a metric space, There is an isometric imbedding of X into a complete metric space.

7.1.12.1 Isometric imbedding of X in Y : an imbedding f:X->Y have the property that for every pair of points x_{1}, x_{2} of X, d_{Y} (f(x_{1}), f(x_{2})) = d_{X} (x_{1}, x_{2})

7.1.12.2 Pf) ( a fixed point : an element of the function’s domain that is mapped to itself by the function)

7.1.13 Completion \bar{h(X)} of X ( a metric space X) : a subspace \bar{h(x)} of Y, if h: X->Y is an isometric imbedding of X into a complete metric space Y

7.1.13.1 Completion of X is uniquely determined up to an isometry.

7.2 Space-filling Curve

7.2.1 (Thm 44.1) Let I = [0,1]. There exists a continuous map f: I -> I^{2} whose image fills up the entire square I^{2}.

7.2.1.1 Pf) (closed -> complete -> \mathcal{C}(I,I^{2}) is complete)

7.3 Compactness in Metric spaces

7.3.1 Every compact metric space is complete

7.3.1.1 Pf) (Lem 43.1)

7.3.2 Totally bounded ( a metric spacx (X,d)) : for every \epsilon >0, there is a finite covering of X by \epsilon -balls.

7.3.3 (Thm 45.1) A metric space (X,d) is compact iff it is complete and totally bounded.

7.3.3.1 Pf) (Every compact metric space is complete)

7.3.4 Equicontinuous \mathcal{F} at x_{0} (a subset \mathcal{F} of the function space \mathcal{C} (X,Y), a x_{0} \in X) : Let (Y,d) be a metric space. If given \epsilon >0, there is a neighborhood U of x_{0} s.t. for all x \in U and all f \in \mathcal{F}, d(f(x), f(x_{0})) < \epsilon .

7.3.4.1 Equicontinuous (a set \mathcal{F}) : the set \mathcal{F} is equicontinuous at x_{0} for each x_{0} \in X.

7.3.5 (Lem 45.2) Let X be a space; Let (Y,d) be a metric space. If the subset \mathcal{F} of \mathcal{C} (X,Y) is totally bounded under the uniform metric corresponding to d, then \mathcal{F} is equicontinuous under d.

7.3.6 (Lem 45.3) Let X be a pace; let (Y,d) be a metric space; assume X and Y are compact. If the subset \mathcal{F} of \mathcal{C}(X,Y0 is equicontinuous under d, then \mathcal{F} is totally bounded under the uniform and sup metrics corresponding to d.

7.3.7 Pointwise bounded \mathcal{F} under d ( a subset \mathcal{F} of the function space \mathcal{C} (X,Y) , a metric space (Y,d)) : for each x \in X, the subset \mathcal{F}_{a} = {f(a) | f \in \mathcal{F}} of Y is bounded under d.

7.3.8 (Ascoli’s Theorem, classical version) (Thm 45.4) : Let X be a compact space; Let (\mathbb{R},d) denote euclidean space in either the square metric or the Euclidean metric; give \mathcal{C} (X,\mathbb{R}^{n}) the corresponding uniform topology. A subspace \mathcal{F} of \mathcal{C} (X,\mathbb{R}^{n}) has compact closure iff \mathcal{F} is equicontinuous and pointwise bounded under d.

7.3.8.1 Pf) (Thm 45.1) (Lem 45.2) (Lem 45.3)

7.3.8.2 (Cor) Let X be compact; let d denote either the square metric or the Euclidean metric on \mathbb{R}^{n} ; give \mathcal{C} (X,\mathbb{R}^{n}) the corresponding uniform topology. A subspace \mathcal{F} of \mathcal{C} (X,\mathbb{R}^{n}) is compact iff it is closed, bounded under the sup metric \rho, and equicontinuous under d.

7.4 Pointwise and Compact convergence

7.4.1 Topology of pointwise convergence : given a point x of the set X and an open set U of space Y, let S(x,U) = {f | f \in Y^{X} and f (x) \in U}. The sets S(x,U) are a subbasis for topology on Y^{X}.

7.4.1.1 Also called ‘point-open topology’

7.4.2 (Thm 46.1) A sequence f_{n} of functions converges to the function f in the topology of pointwise convergence iff for each x in X, the sequence f_{n} (x) of points of Y converges to the point f(x).

7.4.2.1 Pf) (Lem 43.3)

7.4.3 Topology of compact convergence : Let (Y,d) be a metric space; let X be a topological space. Given an element f of Y^{X}, a compact subspace C of X, and a number \epsilon >0, let B_{C} (f, \epsilon) denote the set of all those elements g of Y^{X} for which sup{d(f(x), g(x)) | x \in C} < \epsilon. The sets B_{C} (f, \epsilon ) form a basis for a topology on Y^{X}.

7.4.3.1 Also called ‘Topology of uniform convergence on compact sets’

7.4.4 (Thm 46.2) A sequence f_{n} : X -> Y of functions converges to the function f in the topology of compact convergence iff for each compact subspace C of X, the sequence f_{n} | C converges uniformly to f | C.

7.4.5 Compactly generated (a space X) : it satisfies the following condition.

7.4.5.1 A set A is open in X if A \cap C is open in C for each compact subspace C of X.

7.4.6 (Lem 46.3) If X is locally compact, or if X satisfies the first countability axiom, then X is compactly generated.

7.4.7 (Lem 46.4) If X is compactly generated, then a function f: X->Y is continuous if for each compact subspace C of X, the restricted function f | C is continuous.

7.4.8 (Thm 46.5) Let X be a compactly generated space; let (Y,d) be a metric space. Then \mathcal{C} (X,Y) is closed in Y^{X} in the topology of compact convergence.

7.4.8.1 Pf) ( Uniform limit theorem)

7.4.8.2 (Cor 46.6) Let X be a compactly generated space; let (Y,d) be a metric space. If a sequence of continuous functions f_{n} : X -> Y converges to f in the topology of compact convergence, then f is continuous.

7.4.9 (Thm 46.7) Let X be a space; let (Y,d) be a metric space. For the function space Y^{X}, one has the following inclusions of topologies :

7.4.9.1 (Uniform) \supseteq (Compact convergence) \supseteq (Pointwise convergence)

7.4.9.2 If X is compact, the first two coincide, and if X is discrete, the second two coincide.

7.4.10 Compact-open topology : Let X and Y be topological spaces. If C is a compact subspace of X and U is an open subset of Y, define S(C,U) = {f | f \in \mathcal{C} (X,Y) and f(C) \subset U}. The sets S(C,U) forms a subbasis for a topology on \mathcal{C} (X,Y).

7.4.11 (Thm 46.8) Let X be a space and let (Y,d) be a metric space. On the set \mathcal{C} (X,Y), the compact-open topology and the topology of compact convergence coincide.

7.4.11.1 (Cor 46.9) Let Y be a metric space. The compact convergence topology on \mathcal{C} (X,Y) does not depend on the metric of Y. Therefore if X is compact, the uniform topology on \mathcal{C} (X,Y) does not depend on the metric of Y.

7.4.12 (Thm 46.10) Let X be a locally compact Hausdorff; let \mathcal{C} (X,Y) have the compact-open topology. Then the map e: X \times \mathcal{C} (X,Y) -> Y defined by the equation e(x,f) = f(x) is continuous.

7.4.12.1 Evaluation map : the map e

7.4.13 Map F induced by f ( a function f : X \times Z -> Y) : There is a corresponding function F : Z -> \mathcal{C} (x<Y), defined by the equation (F(z))(x) = f(x,z). Conversely, given F : Z -> \mathcal{C} (X,Y), this equation defines a corresponding function f : X \times Z -> Y.

7.4.14 (Thm 46.11) Let X and Y be spaces; give \mathcal{C} (X,Y) the compact-open topology. If f: X \times Z -> Y is continuous, then so is the induced function F: Z -> \mathcal{C} (X,Y). The converse holds if X is locally compact Hausdorff.

7.4.14.1 Pf) (Tube lemma)

7.4.15 Homotopic (two functions f and g in \mathcal{C} (X,Y) ) : There is a continuous map h : X \times [0,1] -> Y s.t. h(x,0) = f(x) and h(x,1) = g(x) for each x \in X.

7.4.15.1 Homotopy between f and g : the map h

7.5 Ascoli’s Theorem

7.5.1 (Ascoli’s Theorem) (Thm 47.1) : Let X be a space and let (Y,d) be a metric space. Give \mathcal{C} (X,Y) the topology of compact convergence; Let \mathcal{F} be a subset of \mathcal{C} (X,Y).

7.5.1.1 If \mathcal{F} is equicontinuous under d and the set \mathcal{F}_{a} = { f(a) | f\in \mathcal{F} } has compact closure for each a \in X, then \mathcal{F} is contained in a compact subspace of \mathcal{C} (X,Y).

7.5.1.2 The converse holds if X is locally compact Hausdorff.

7.5.1.3 Pf) (Tychonoff Theorem) (Thm 46.8) (Thm 46.10) (Thm 45.1) (Lem 45.2)

Baire Spaces and Dimension Theory

8 Baire Spaces and Dimension Theory

8.1 Baire Spaces

8.1.1 A has Empty interior ( a subset A of a space X) : A contains no open set of X other than the empty set.

8.1.1.1 A has empty interior if every point of A is a limit point of the complement of A.

8.1.1.2 The complement of A is dense in X.

8.1.2 Baire space ( a space X) : if the following condition holds.

8.1.2.1 Given any countable collection {A_{n}} of closed sets of X each of which has empty interior in X, their union \bigcup A_{n} also has empty interior in X.

8.1.3 First category in X ( a subset A of a space X) : it was contained in the union of a countable collection of closed sets of X having empty interiors in X.

8.1.3.1 Second category in X (a subset A of a space X) : A is not on the first category in X.

8.1.3.2 A space X is a Baire space iff every nonempty open set in X is of the second category.

8.1.4 (Lem 48.1) X is a Baire space iff given any countable collection {U_{n}} of open sets in X, each of which is dense in X, their intersection \bigcap U_{n} is also dense in X.

8.1.5 (Baire Category theorem) (Thm 48.2) : If X is a compact Hausdorff space or a complete metric space, then X is a Baire space.

8.1.6 (Lem 48.3) Let C_{1} \supset C_{2} \supset … be a nested sequence of nonempty closed sets in the complete metric space X. If diam C_{n} -> 0, then \bigcap C_{n} \neq \empty

8.1.7 (Lem 48.4) Any open subspace Y of a Baire space X is itself a Baire space.

8.1.8 (Thm 48.5) Let X be a space; let (Y,d) be a metric space. Let f_{n} : X -> Y be a sequence of continuous functions s.t. f_{n} (x) -> f(x) for all x \in X, where f : X-> Y. If X is a baire space, the set of points at which f is continuous is dense in X

8.1.8.1 Pf) (Lem 48.4)

8.2 Nowhere-Differentiable function

8.2.1 (Thm 49.1) Let h : [0,1] -> \mathbb{R} be a continuous function. Given \epsilon >0, there is a function g : [0,1] -> \mathbb{R} with \vert h(x) -> g(x) \vert < \epsilon for all x, s.t. g is continuous and nowhere differentiable.

8.2.1.1 Pf) (Lemma 48.1) (Complete metric space -> Baire space)

8.2.1.1.1 piecewise-linear (a function g) : a function whose graph is a broken line segment; each line segment in the graph of g has slope at least \alpha in absolute value

8.2.1.1.2 sawtooth graph

8.3 Introduction to Dimension Theory

8.3.1 \mathcal{A} has order m+1 ( a collection \mathcal{A} of subsets of the space X) : some points of X lies in m+1 elements of \mathcal{A}, and no point of X lies in more than m+1 elements of \mathcal{A}.

8.3.2 Finite dimensional ( a space X) : There is some integer m s.t. for every open covering \mathcal{A} of X, there is an open covering \mathcal{B} of X that refines \mathcal{A} and has order at most m+1.

8.3.3 Topological dimension of X (a space X) : the smallest value of m for a space X to be finite dimensional.

8.3.3.1 Dim X : Topological dimension of X

8.3.3.2 Example

8.3.3.2.1 Any compact subspace X of \mathcal{R} has topological dimension at most 1.

8.3.3.2.2 Any compact subspace X of \mathcal{R}^{2} has topological dimension at most 2.

8.3.4 (Thm 50.1) Let X be a space having finite dimension. If Y is a closed subspace of X, then Y has finite dimension and dim Y \le dim X.

8.3.5 (Thm 50.2) Let X = Y \cup Z, where Y and Z are closed subspaces of X having finite topological dimension. Then dim X = max {dim Y, dim Z}

8.3.5.1 Pf) (Thm 50.1)

8.3.5.2 (Cor 50.3) Let X = Y_{1} \cup … \cup Y_{k}, where each Y_{i} is a closed subspace of X and is finite dimensional. Then dim X = max{dim Y_{1} , .., dim Y_{k}}.

8.3.6 Arc : a space homeomorphic to the closed unit interval

8.3.6.1 Endpoints of Arc : the points p and q s.t. A – {p} and A-{q} are connected.

8.3.6.2 Linear graph G : a Hausdorff space that is written as the union of finitely many arcs, each pair of which intersects in at most a common end point.

8.3.6.3 Edges of G : arcs in the collection

8.3.6.4 Vertices of G : endpoints of arcs

8.3.6.5 Each edge of G is closed in G; G has topological dimension 1.

8.3.6.6 Every finite linear graph can be imbedded in \mathbb{R}^{3}

8.3.7 Geometrically independent (a set {\mathbf{x}{0},…, \mathbf{x}{k}} of points of \mathbb{R}^{N} ) : if the equation \sum_{I = 0}^{k} a_{i}mathbf{x}{i} = \mathbf{0} and \sum{I = 0}^{k} a_{i} = 0 holds only if each a_{i} = 0.

8.3.8 Plane P determined by these points (a set of all points \mathbf{x} of of \mathbb{R}^{N} ) : Let {\mathbf{x}{0},…, \mathbf{x}{k}} be a set of points of \mathbb{R}^{N} that is geometrically independent.\mathbf{x} = \sum_{I = 0}^{k} t_{i}mathbf{x}{i} where \sum{I = 0}^{k} t_{i} = 1.

8.3.8.1 mathbf{x} = \mathbf{x}{0} + \sum{I = 0}^{k} a_{i}( mathbf{x}{i} - \mathbf{x}{0}) for some scalars a_{1} , …, a_{k}

8.3.8.2 Translation T of \mathbb{R}^{N} : a homeomorphism T : \mathbb{R}^{N} -> \mathbb{R}^{N} ; T(\mathbf{x}) = \mathbf{x – x_{0}}

8.3.8.3 K-plane P in \mathbb{R}^{N} : a plane P

8.3.9 A in general position in \mathbb{R}^{N} (a set A of points of \mathbb{R}^{N}) : every subset of A containing N + 1 or fewer points is geometrically independent

8.3.10 (Lem 50.4) Given a finite set {\mathbf{x}{1},…, \mathbf{x}{n}} of points of \mathbb{R}^{N} and given \delta >0, there exists a set {\mathbf{y}{1},…, \mathbf{y}{n}} of points of \mathbb{R}^{N} in general position in \mathbb{R}^{N}, s.t. \vert x_{i} – y_{i} \vert < \delta for all i.

8.3.10.1 Pf) (Def of Baire space)

8.3.11 (The imbedding Theorem) (Thm 50.5) Every compact metrizable space X of topological dimension m can be imbedded in \mathbb{R}^{2m+1}

8.3.12 (Thm 50.6) Every compact subspace of \mathbb{R}^{N} has topological dimension at most N.

8.3.12.1 Pf) M-cube : homeomorphic to the product (0,1)^{M}

8.3.12.2 (Cor 50.7) Every compact m-manifold has topological dimension at most m.

8.3.12.3 (Cor 50.8) Every compact m-manifold con be imbedded in \mathbb{R}^{2m+1}

8.3.12.4 (Cor 50.9) Let X be a compact metrizable space. Then X can be imbedded in some Euclidean space \mathbb{R}^{N} iff X has finite topological dimension.

Fundamental Group

9 Fundamental Group

9.1 Homotopy of paths

9.1.1 f is homotopic to f’ (two continuous maps f, f’ of the space X into the space Y) : Let I = [0,1]. there is a continuous map F : X \times I -> Y s.t. F(x,0) = f(x) and F(x,1) = f’(x).

9.1.1.1 Homotopy between f and f’ : the map F

9.1.1.2 f \simeq f’ : f is homotopic to f’

9.1.1.3 nulhomotopic f : If f \simeq f’ and f’ is a constant map

9.1.2 Path in X from x_{0} to x_{1} ( a continuous map f : [0,1] -> X ) : f(0) = x_{0} , f(1) = x_{1}

9.1.2.1 Initial point : x_{0}

9.1.2.2 Final point : x_[1]

9.1.3 Path homotopic ( two paths f, f’ : [0,1] ->X ) : They have the same initial point x_{0} and the same final point x_{1} and there is a continuous map F : I \times I -> X for each s \in I and each t \in I s.t.

9.1.3.1 F (s, 0) = f(s) and F (s, 1) = f’(s)

9.1.3.2 F (0,t) = x_{0} and F (1, t) = x_{1}

9.1.3.2.1 F : a path homotopy between f and f’

9.1.3.2.2 f \simeq_{p} f’ : f is path homotopic to f’

9.1.4 (Lem 51.1) The relations \simeq and \simeq_{p} are equivalence relations.

9.1.5 [ f ] : path-homotopy equivalence class of a path f

9.1.6 Example

9.1.6.1 Straight-line homotopy ( two maps f, g X -> \mathbb{R}^{2}) : F(x,t) = (1-t) f(x) + t g(x)

9.1.6.2 Let A be any convex subspace of \mathbb{R}^{n}. Then any two paths f, g in A from x_{0} to x_{1} are path homotopic in A.

9.1.7 Product f * g of f and g ( a path f in X from x_{0} to x_{1}, a path g in X from x_{1} to x_{2} ) : a path h given by the equations

9.1.7.1 h(s) = \begin{cases} f(2s) for s \in [0,frac{1}{2}] , \ g(2s – 1) for s \in [frac{1}{2}, 1] \end{cases}

9.1.7.2 The function h is well-defined and continuous.

9.1.7.2.1 Pf) pasting lemma

9.1.7.3 h is a path in X from x_{0} to x_{2}

9.1.8 (Groupoid properties of * )(Thm 51.2) The operation * has the following properties.

9.1.8.1 (Associativity) If [ f ] * ( [ g ] * [ h ] ) is defined, so is ( [ f ] * [ g ] ) * [ h ] ,and they are equal.

9.1.8.2 (Right and left identities) Given x \in X, let e_{x} denote the constant path e_{x} : I -> X carrying all of I to the point x. If f is a path in X from x_{0} to x_{1}, then [ f ] * [ e_{x_{1}} ] = [ f ] and [ e_{x_{0}} ] * [ f ] = [ f ]

9.1.8.3 (Inverse) Given the path f in X from x_{0} to x_{1}, let \bar{f} be the path defined by \bar{f} (s) = f (1-s) . It is called the reverse of f. Then [ f ] * [ \bar{f} ] = [ e_{x_{0}} ] and [ \bar{f} ] * [ f ] = [ e_{x_{1}} ]

9.1.8.4 Pf) convex

9.1.9 (Thm 51.3) Let f be a path in X, and let a_{0}, …, a_{n} be numbers s.t. 0 = a_{0} < a_{1} < … < a_{n} = 1. Let f_{i} : I -> X be the path that equals the positive linear map of I onto [a_{i-1}, a_{i}] followed by f. Then [ f ] = [ f_{1} ] * .. * [ f_{n} ]

9.2 Fundamental Group

9.2.1 Suppose G and G’ are groups, written multiplicatively.

9.2.1.1 Homomorphism (a map f : G -> G’) : f (x \cdot y) = f (x) \cdot f (y).

9.2.1.1.1 f( e ) = e’

9.2.1.1.2 f ( x^{-1} ) = f ( x ) ^ {-1}

9.2.1.1.3 Kernel of f : the set f^{-1} (e’) , which is subgroup of G

9.2.1.1.4 Monomorphism : an injective homomorphism

9.2.1.1.5 Epimorphism : a surjective homomorphism

9.2.1.1.6 Isomorphism : a bijective homomorphism

9.2.1.2 Left coset xH of H in G (a subgroup H of a group G ) : the set of all products xh , for h \in H

9.2.1.3 Normal subgroup H of G (a subgroup H of a group G ) : x \cdots h \cdots x^{-1} \in H for each x \in G and each h \in H.

9.2.1.4 G/H : a partition which is the collection of all coset forms.

9.2.2 Quotient of G by H (a subgroup H of a group G ) : a group (G/H, \cdot ) which \cdot is a well-defined operation (x H) \cdot (y H) = (x \cdot y) H.

9.2.2.1 If f : G -> G’ is an epimorphism, its kernel N is a normal subgroup of G and f induces an isomorphism G/N -> G’ ; x N -> f(x) for each x \in G.

9.2.3 Loop based at x_{0} ( a point x_{0} of a space X ) : a path in X that begins and ends at x_{0}

9.2.4 Fundamental group of X relative to the base point x_{0} ( a point x_{0} of a space X ) : The set of path homotopy classes of loops based at x_{0}

9.2.4.1 \pi_{1} (X, x_{0}) <- Fundamental group of X relative to the base point x_{0}

9.2.4.2 The operation * restricted to this set satisfies axioms for a group.

9.2.4.3 Also called ‘ First homotopy group of X ‘

9.2.4.4 Example

9.2.4.4.1 For a convex subset X of \mathbb{R}^{n} \pi_{1} (X, x_{0}) is trivial group.

9.2.4.4.2 Unit ball B^{n} in \mathbb{R}^{n} has trivial fundamental group.

9.2.5 \alpha – hat : let \alpha be a path in X from x_{0} to x_{1}. A map \alpha – hat is \hat{\alpha} : \pi_{1} (X, x_{0}) -> \pi_{1} (X, x_{1}) by the equation \hat{\alpha}([ f ]) – [ \bar{\alpha} ] * [ f ] * [ \alpha ].

9.2.6 (Thm 52.1) The map \hat{\alpha} is a group isomorphism.

9.2.6.1 (Cor 52.2) If X is path connected and x_{0} and x_{1} are two points of X, then \pi_{1} (X, x_{0}) is isomorphic to \pi_{1} (X, x_{1})

9.2.7 Simply connected ( a space X) : X is a path-connected space and if \pi_{1} (X, x_{0}) is the trivial (one-element) group for some x_{0} \in X, and hence for every x_{0} \in X.

9.2.7.1 \pi_{1} (X, x_{0}) : \pi_{1} (X, x_{0}) is the trivial group.

9.2.8 (Lem 52.3) In a simply connected space X, any two paths having the same initial and final points are path homotopic.

9.2.9 h : (X, x_{0}) -> (Y, y_{0}) <- a continuous map that carries the point x_{0} of X to the point y_{0} of Y

9.2.10 Homomorphism h_{} induced by h relative to the base point x_{0} (a continuous map h : (X, x_{0}) -> (Y, y_{0}) ) : h_{} : \pi_{1} (X, x_{0}) -> \pi_{1} (Y, y_{0}) by the equation h_{*} ([ f ]) = [h \bullet f ]

9.2.11 (Thm 52.4) If h : (X, x_{0}) -> (Y, y_{0}) and k : (Y, y_{0}) -> (Z, z_{0}) are continuous, then (k \bullet h){*} = k{} \bullet h_{} . If i: (X, x_{0}) -> (X, x_{0}) is the identity map, then i_{*} is the identity homomorphism.

9.2.11.1 (Cor) If h : (X, x_{0}) -> (Y, y_{0}) is a homomorphism of X with Y, then h_{*} is an isomorphism of \pi_{1} (X, x_{0}) with \pi_{1} (Y, y_{0})

9.3 Covering Spaces

9.3.1 Evenly covered by p ( an open set U of B, a map p : E -> B ) : Let p be continuous surjective map. The inverse image p^{-1} (U) can be written as the union of disjoint open sets V_{\alpha} in E s.t. for each \alpha, the restriction p to V_{\alpha} is a homeomorphism of V_{\alpha} onto U.

9.3.1.1 A partition of p^{-1} (U) into slices : the collection {V_{\alpha}}

9.3.2 Covering map ( a continuous surjective map p : E -> B) : Every point b of B has a neighborhood U that is evenly covered by p.

9.3.2.1 a covering space E of B : if a covering map p : E -> B exists

9.3.2.2 If p : E->B is a covering map, then for each b \in B the subspace p^{-1}(b) of E has the discrete topology.

9.3.2.3 If p : E ->B is a covering map, then p is an open map.

9.3.3 (Thm 53.1) The map p : \mathbb{R} -> S^{-1} given by the equation p(x) = (cos 2 \pi x , sin 2 \pi x ) is a covering map.

9.3.4 A local homeomorphism p of E with B ( a map p : E-> B) : each point e of E has a neighborhood that is mapped homeomorphically by p onto an open subset of B.

9.3.4.1 If p: E -> B is a covering map, then p is a local homeomorphism of E with B.

9.3.4.2 Example

9.3.4.2.1 The map p : \mathbb{R}_{+} -> S^{1} ; p(x) = (cos 2 \pi x , sin 2 \pi x )

9.3.5 (Thm 53.2) : Let p : E->B be a covering map. If B_{0} is a subspace of B, and if E_{0} = p^{-1}(B_{0}), then the map p_{0} : E_{0} -> B_{0} obtained by restricting p is a covering map.

9.3.6 (Thm 53.3) If p : E -> B and p’ : E’ -> B’ are covering maps, then p \times p’ : E \times E’ -> B \times B’ is a covering map.

9.3.6.1 Example

9.3.6.1.1 For a Torus T = S^{-1} \times S^{-1}, p \times p : \mathbb{R} \times \mathbb{R} -> T

9.3.6.1.2 Let b_{0} = p(0) of S^{1}; a figure-eight space B_{0} = a subspace (S^{1} \times b_{0}) U (b_{0} \times S^{1}) of S^{1} \times S^{1} ; an infinite grid E_{0} = (\mathbb{R} \times \mathbb{Z}) \cup (\mathbb{Z} \times \mathbb{R} ). Then the map p_{0} : E_{0} -> B_{0} = p \times p | E_{0} is a covering map.

9.4 Fundamental group of the circle

9.4.1 Lifting of f ( a continuous mapping f of some space X into B) : Let p : E-.B be a map. A lifting of f is a map \tilde{f} : X -> E s.t. p \bullet \tilde{f} = f.

9.4.2 (Lem 54.1) Let p : E -> B be a covering map, let p(e_{0}) = b_{0}. Any path f : [0,1] -> B beginning at b_{0} has a unique lifting to a path \bar{f} in E beginning at e_{0}.

9.4.2.1 Pf) (Lebesgue number lemma) (pasting lemma)

9.4.3 (Lem 54.2) Let p : E->B be a covering map; let p(e_{0}) = b_{0}. Let the map F : I \times I -> B be continuous, with F (0,0) = b_{0}. There is a unique lifting of F to a continuous map \tilde{F} : I \times I -> E s.t. \tilde{F} (0,0) = e_{0}. If F is a path homotopy, then \tilde{F} is a path homotopy.

9.4.3.1 Pf) (Lem 54.1) (Lebesgue number lemma) (pasting lemma)

9.4.4 (Thm 54.3) Let p : E -> B be a covering map ; let p(e_{0}) = b_{0}. Let f and g be two paths in B from b_{0} to b_{1}; let \tilde{f} and \tilde{g} be their respective liftings to paths in E beginning ar e_{0}. If f and g are path homotopic, then \tilde{f} and \tilde{g} end at the same point of E and path homotopic.

9.4.5 Lifting correspondence \phi derived from the covering map p ( a covering map p: E -> B) : Let b_{0} \in B. Choose e_{0} so that p(e_{0}} = b_{0}. Given an element [ f ] of \pi_{1} (B,b_{0}) , let \tilde{F} be the lifting of f to a path in E that begins at e_{0}. Let \phi([ f ]) denote the end point \tilde{f} (1) of \tilde{f}. Then \phi is a well-defined set map \phi : \pi_{1} (B,b_{0}) -> p^{-1} (b_{0}).

9.4.6 (Thm 54.4) Let p: E->B be a covering map; let p(e_{0}) = b_{0}. If E is a path connected, then the lifting correspondence \phi : \pi_{1} (B,b_{0}) -> p^{-1} (b_{0}). Is surjective. If E is simply connected, it is bijective.

9.4.7 (Thm 54.5) The fundamental group of S^{1} is isomorphic to the additive group of integers.

9.4.7.1 Pf) (Thm 53.1)

9.4.8 Cyclic group (a group G) : Let x be an element of G. the set of all elements of the form x^{m}, for m \in \mathbb{Z}, equals G

9.4.8.1 The symbol x^{n} denotes the n-fold product of x with itself, x^{-n} denotes the n-fold product of x^{-1} with itself, and x^{0} denotes the identity element of G.

9.4.8.2 x : a generator of G

9.4.9 Order of the group : Cardinality of a group

9.4.10 (Thm 54.6) Let p : E->B be a covering map ; let p(e_{0}) = b_{0}.

9.4.10.1 The homomorphism p_{*} : \pi_{1} (E,e_{0}) -> \pi_{1} (B,b_{0}) is a monomorphism.

9.4.10.2 Let H = p_{*}( \pi_{1} (E,e_{0}) ). The lifting correspondence \phi induces an injective map \Phi : \pi_{1} (B,b_{0}) / H -> p^{-1} (b_{0}) of the collection of right cosets of H into p^{-1} (b_{0}), which is bijective if E is path connected.

9.4.10.3 If f is a loop in B based at b_{0}, then [ f ] \in H iff f lifts to a loop in E based at e_{0}.

9.4.10.3.1 Pf) \phi ( [ f ] ) = \phi ( [ g ] ) iff [ f ] \in H * [ g ].

9.5 Retractions and Fixed Points

9.5.1 Retraction of X onto A ( a subset A of X ) : a continuous map r : X -> A s.t. r |A is the identity map of A.

9.5.1.1 Retract A of X : If such a map r exists

9.5.2 (Lem 55.1) If A is a retract of X, then the homomorphism of fundamental groups induced by inclusion j : A ->X is injective.

9.5.3 (No-retraction theorem) (Thm 55.2) : There is no retraction of B^{2} onto S^{1}.

9.5.3.1 B^{2} is a unit ball.

9.5.4 (Lem 55.3) Let h : S^{1} -> X be a continuous map. Then the following conditions are equivalent.

9.5.4.1 h is nullhomotopic.

9.5.4.2 h extends to a continuous map k : B^{2} -> X.

9.5.4.3 h_{*} is the trivial homomorphism of fundamental groups.

9.5.5 (Cor 55.4) The inclusion map j : S^{1} -> \mathbb{R}^{2} - \mathbf{0} is not nullhomotopic. The identity map I : S^{1} -> S^{1} is not nullhomotopic.

9.5.6 (Thm 55.5) Given a nonvanishing vector field on B^{2}, there exists a point of S^{1} where the vector field points directly inward and a point of s^{1} where it points directly outward.

9.5.6.1 Vector field on B^{2} : an ordered pair (x, v(x)), where x is in B^{2} and v is a continuous map of B^{2} into \mathbb{R}^{2}.

9.5.6.2 Nonvanishing (a vector field) : v(x) \neq \mathbf{0} for every x

9.5.6.3 Pf) (Cor 55.4)

9.5.7 (Brower fixed-point theorem for the disc) (Thm 55.6) : If f : B^{2} -> B^{2} is continuous, then there exists a point x \in B^{2} s.t. f(x) = x

9.5.7.1 (Cor 55.7) Let A be a 3 by 3 matrix of positive real numbers. Then A has a positive real eigenvalue.

9.5.8 (Thm 55.8) There is an \epsilon > 0 s.t. for every open covering \mathcal{A} of T by sets of diameter less than \epsilon, some point of T belongs to at least three elements of \mathcal{A}.

9.5.8.1 T = {(x,y) | x \ge 0 and y \ge 0 and x + y \le 1}

9.5.8.2 Pf) T is homeomorphic to B^{2}, a partition of unity,

9.6 Fundamental Theorem of Algebra

9.6.1 (Fundamental theorem of algebra) (Thm 56.1) A polynomial equation x^{n} + a_{n-1} x^{n-1} + … + a_{1}x + a_{0} = 0 of degree n > 0 with real or complex coefficients has at least one (real or complex) root.

9.6.1.1 Pf) g: S^{1} -> \mathbb{R}^{2} - \mathbf{0} ; g(z) = z^{n} is not nullhomotopic.

9.7 Borsuk-Ulam Theorem

9.7.1 Antipode (a point x in S^{n}) : -x

9.7.2 Antipode-preserving h ( a map h : S^{n} -> S^{m}) : h(-x) = -h(x) for all x \in S^{n}.

9.7.3 (Thm 57.1) if h : S^{1} -> S^{1} is continuous and antipode-preserving, then h is not nullhomotopic.

9.7.3.1 Pf) covering map,

9.7.4 (Thm 57.2) There is no continuous antipode-preserving map g : S^{2} -> S^{1}.

9.7.4.1 Pf) (Thm 57.1)

9.7.5 (Borsuk-Ulam theorem for S^{2}) (Thm 57.3) Given a continuous map f : S^{2} -> \mathbb{R}^{2}, there is a point x of S^{2} s.t. f(x) = f(-x).

9.7.6 (The bisection theorem) (Thm 57.4) Given two bounded polygonal regions in \mathbb{R}^{2}, there exists a line in \mathbb{R}^{2} that bisects each of them.

9.7.6.1 Pf) (Borsuk-Ulam theorem)

9.8 Deformation Retracts and Homotopy type

9.8.1 (Lem 58.1) Let h, k : (X, x_{0}) -> (Y, y_{0}) be continuous maps, If h and k are homotopic, and if the image of the base point x_{0} of X remains fixed at y_{0} during the homotopy, then the homomorphisms h_{} and k_{} are equal.

9.8.2 (Thm 58.2) The inclusion map j : S^{n} -> \mathbb{R}^{n+1} - \mathbf{0} induces an isomorphism of fundamental groups.

9.8.2.1 Pf) (Lem 58.1)

9.8.3 Deformation retract of X ( a subspace A of X) : the identity map of X is homotopic to a map that carries all of X into A, such that each point of A remains fixed during the homotopy.

9.8.3.1 Deformation retraction H of X onto A ( a homotopy H) : There is a continuous map H : X \times I -> X s.t. H(x,0) = x and H(x,1) \in A for all x \in X, and H(a,t) = a for all a \in A.

9.8.3.2 The map r : X -> A defined by the equation r(x) = H(x,1) is a retraction of X onto A.

9.8.3.3 H is a homotopy between the identity map of X and the map j \bullet r, where j : A -> X is inclusion.

9.8.4 (Thm 58.3) Let A be a deformation retract of X ; let x_{0} \in A. Then the inclusion map j : (A, x_{0}) -> (X, x_{0}) induces an isomorphism of fundamental groups.

9.8.5 Homotopy equivalences ( continuous maps f : X -> Y, g : Y -> X) : the map g \bullet f : X -> X is homotopic to the identity map of X, and the map f \bullet g : Y -> Y is homotopic to the identity map of Y.

9.8.5.1 Each is said to be a ‘homotopy inverse’ of the other.

9.8.5.2 Relation of homotopy equivalence is an equivalence relation.

9.8.5.3 Same Homotopy type : Spaces that are homotopy equivalent

9.8.6 (Lem 58.4) Let h, k : X -> Y be continuous maps; let h(x_{0}) = y_{0} and k(x_{0}) = y_{1}. If h and k are homotopic, there is a path \alpha in Y from y_{0} to y_{1} s.t. k_{} = \hat{\alpha} \bullet h_{}. Indeed, if H : X \times I -> Y is the homotopy between h and k, then \alpha is the path \alpha(t) = H(x_{0}, t)

9.8.6.1 (Cor 58.5) Let h, k : X ->Y be homotopic continuous maps; let h(x_{0}) = y_{0} and k(x_{0}) = y_{1}. If h_{} is injective, or surjective, or trivial, so is k_{}.

9.8.6.2 (Cor 58.6) Let h : X -> Y. If h is nullhomotopic, then h_{*} is the trivial homomorphism.

9.8.7 (Thm 58.7) Let f : X -> Y be continuous; let f(x_{0}) = y_{0}. If f is a homotopy equivalence, then f_{*} : \pi_{1} (X, x_{0}) ->\pi_{1} (Y, y_{0}) is an isomorphism.

9.9 Fundamental Group of S^{n}

9.9.1 (Thm 59.1) Suppose X = U \cup V, where U and V are open sets of X. Suppose that U \cap V is path connected, and that x_{0} \in U \cap V. Let I and j be the inclusion mappings of U and V, respectively, into X. Then the images of the induced homomorphisms i_{} : \pi_{1} (U, x_{0}) ->\pi_{1} (X, x_{0}) and j_{} : \pi_{1} (V, x_{0}) ->\pi_{1} (X, x_{0}) generate \pi_{1} (X, x_{0})

9.9.1.1 Pf) there is a subdivision a_{0} < a_{1} < … < a_{n} of the unit interval s.t. f(a_{i}) \in U \cap V and f([a_{i-1}, a_{i}]) is contained either in U or in V, for each i., (Lebesgue number lemma)

9.9.1.2 (Cor 59.2 ) Suppose X = U \cup V, where U and V are open sets of X; suppose X \cap V is nonempty and path connected, If U and V are simply connected, then X is simply connected.

9.9.2 (Thm 59.3) If n \ge 2, then n-sphere S^{n} is simply connected.

9.9.2.1 Stereographic projection f : (S^{n} – p) -> \mathbb{R}^{n} ; f(x) = f(x_{1}, …, x_{n+1}) = frac{1}{1-x_{n+1}} (x_{1}, …, x_{n}) where p = (0,…,0,1) \in \mathbb{R}^{n+1}

9.9.2.2 Pf) (Cor 59.2)

9.10 Fundamental Groups of Some Surfaces

9.10.1 (Thm 60.1) \pi_{1} (X \times Y, x_{0} \times y_{0}) is isomorphic with \pi_{1}(X,x_{0}) \times \pi_{1} (Y, y_{0}).

9.10.1.1 (Cor 60.2) The fundamental group of the torus T = S^{1} \times S^{1} is isomorphic to the group \mathbb{Z} \times \mathbb{Z}.

9.10.2 Projective plane P^{2} : quotient space obtained from S^{2} by identifying each point x of S^{2} with its antipodal point -x.

9.10.3 (Thm 60.3) The projective plane P^{2} is a compact surface, and the quotient map p: S^{2} -> P^{2} is a covering map.

9.10.3.1 Pf) S^{2} is normal and p is a closed map -> P^{2} is Hausdorff

9.10.3.2 (Cor 60.4) \pi_{1} (P^{2}, y) is a group of order 2.

9.10.3.2.1 Pf) (Thm 54.4)

9.10.4 Projective n-space : quotient space obtained from S^{n} by identifying each point x of S^{2} with its antipodal point -x

9.10.4.1 The projective plane P^{n} is a compact surface, and the quotient map p: S^{n} -> P^{n} is a covering map.

9.10.5 (Lem 60.5) The fundamental group of the figure eight is not abelian.

9.10.6 (Thm 60.6) The fundamental group of the double torus is not abelian.

9.10.6.1 Double torus T#T : the surface obtained by taking two copies of the torus, deleting a small open disc from each of them, and pasting the remaining pieces together along their edges.

9.10.6.2 Pf) Figure eight X is a retract of T#T.

9.10.6.3 (Cor 60.7) The 2-sphere, torus, projective plane and double torus are topologically distinct.

Separation Theorems in the Plane

10 Separation Theorems in the Plane

10.1 Jordan Separation Theorem

10.1.1 (Lem 61.1) Let C be a compact subspace of S^{2}; let b be a point of S^{2} – C; and let h be a homeomorphism of S^{2} – b with \mathbb{R}^{2}. Suppose U is a component of S^{2} -C. If U does not contain b, then h(U) is a bounded component of \mathbb{R}^{2} – h(C). If U contains b, then h(U-b) is the unbounded component of \mathbb{R}^{2} – h(C).

10.1.2 (Nulhomotopy lemma) (Lem 61.2) : Let a and b be points of S^{2}. Let A be a compact space, and let f: A->S^{2} – a – b be a continuous map. If a and b lie in the same component of S^{2} – f(A), then f is nulhomotopic.

10.1.3 A separates X ( a subset A of a connected space X ) : X – A is not connected

10.1.4 A separates X into n components ( a subset A of a connected space X ): X-A has n components

10.1.5 Simple closed curve : a space homeomorphic to the unit circle S^{1}.

10.1.6 (Jordan separation theorem) (Thm 61.3) : Let C be a simple closed curve in S^{2}. Then C separates S^{2}.

10.1.6.1 Pf) (Thm 59.1)(Thm 55.3) (Nulhomotopy lemma)

10.1.7 (General Separation theorem) (Thm 61.4) Let A_{1} and A_{2} be closed connected subsets of S^{2} whose intersection consists of precisely two points a and b. Then the set C = A_{1} \cup A_{2} separates S^{2}.

10.2 Invariance of Domain

10.2.1 (Homotopy extension lemma) (Lem 62.1) Let X be a space such that X \times I is normal. Let A be a closed subspace of X, and let f : A->Y be a continuous map, where Y is an open subspace of \mathbb{R}^{n}. If f is nulhomotopic, then f may be extended a continuous map g : X ->Y that is also nulhomotopic.

10.2.1.1 Pf) (Tietze Extension theorem)

10.2.2 (Borsuk lemma) (Lem 62.2) Let a and b be points of S^{2}. Let A be a compact space, and let f : A -> S^{2} – a – b be a continuous injective map. If f is nulhomotopic, then a and b lie in the same component of S^{2} – f(A).

10.2.2.1 Pf) (preceding lemma) Let A be a compact subspace of \mathbb{R} ^{2} - \mathbf{0}. If the inclusion j : A -> \mathbb{R} ^{2} - \mathbf{0} is nulhomotopic, then \mathbf{0} lies in the unbounded component of \mathbb{R} ^{2} – A.

10.2.3 (Invariance of domain) (Thm 62.3) If U is an open subset of \mathbb{R} ^{2} and f: U-> \mathbb{R} ^{2} is continuous and injective, then f(U) is open in \mathbb{R} ^{2} and the inverse function f^{-1} : f(U) -> U is continuous.

10.2.3.1 Pf) (Borsuk lemma)

10.3 Jordan Curve theorem

10.3.1 (Thm 63.1) Let X be the union of two open sets U abd V, such that U \cap V can be written as the union of two disjoint open sets A and B. Assume that there is a path \alpha in U from a point a of A to a point b of B, and that there is a path \beta in V from b to a. Let f be the loop f = \alpha * \beta.

10.3.1.1 The path-homotopy class [ f ] generates an infinite cyclic subgroup of \pi_{1} (X,a).

10.3.1.2 If \pi_{1} (X,a) is itself infinite cyclic, it is generated by [ f].

10.3.1.3 Assume there is a path \gamma in U from a to the point a’ of A, and that there is a path \delta in V from a’ to a. Let g be the loop g = \gamma * \delta. Then the subgroups of \pi_{1} (X,a) generated by [ f ] and [ g ] intersect in the identity element alone.

10.3.2 (Nonseparation theorem) (Thm 63.2) Let D be an arc in S^{2}. Then D does not separate S^{2}.

10.3.2.1 Pf) (Borsuk lemma) (Thm 63.1)

10.3.3 (General Nonseparation theorem) (Thm 63.3) Let D_{1} and D_{2} be closed subsets of S^{2} s.t. S^{2} – D_{1} \cap D_{2} is simply connected. If neither D_{1} nor D_{2} separates S^{2}, then D_{1} \cup D_{2} does not separate S^{2}.

10.3.4 (Jordan Curve theorem) (Thm 63.4) Let C be a simple closed curve in S^{2}. Then C separates S^{2} into precisely two components W_{1} and W_{2}. Each of the sets W_{1} and W_{2} has C as its boundary; that is C = \bar{W_{i}} – W_{i} for i = 1,2.

10.3.4.1 Pf) (Jordan separation theorem) (thm 63.1)

10.3.5 (Thm 63.5) Let C_{1} and C_{2} be closed connected subsets of S^{2} whose intersection consists of two points. If neither C_{1} nor C_{2} separates S^{2}, then C_{1} \cup C_{2} separates S^{2} into precisely two componenets.

10.3.6 (Schoenflies theorem) : If C is a simple closed curve in S^{2} and U and V are the components of S^{2} – C, then \bar{U} and \bar{V} are each homeomorphic to the closed unit ball B^{2}.

10.4 Imbedding graphs in the plane

10.4.1 Linear graph G : a Hausdorff space that is written as the union of finitely many arcs.

10.4.1.1 Complete graph on n vertices : G contains exactly n vertices, and if for every pair of distinct vertices of G there is an edge of G joining them

10.4.2 Theta space : a Hausdorff space that is written as the union of three arcs A, B, and C, each pair of which intersects precisely in their end points.

10.4.3 (Lem 64.1) Let X be a theta space that is a subspace of S^{2}; let A, B and C be the arcs whose union is X. Then X separates S^{2} into three components, whose boundaries are A\cup B , B \cup C, and A \cup C, respectively. The component having A \cup B as its boundary equals one of the components of S^{2} – A \cup B.

10.4.3.1 Pf) (Thm 63.5)

10.4.4 (Thm 64.2) Let X be the utilities graph. Then X cannot be imbedded in the plane.

10.4.4.1 Utilities graph : given three houses h_{1}, h_{2}, h_{3}, and three utilities, g (gas), w(water), and e(electricity), Can you connect each utility to each house without letting any of the connecting lines cross?

10.4.5 (Lem 64.3) Let X be a subspace of S^{2} that is a complete graph on four vertices a_{1}, a_{2} , a_{3} and a_{4}. Then X separates S^{2} into four components. The boundaries of these components are the sets X_{1}, X_{2}, X_{3} and X_{4}, where X_{i} is the union of those edges of X that do not have a_{i} as a vertex.

10.4.6 (Thm 64.4) The complete graph on five vertices cannot be imbedded in the plane.

10.5 Winding number of a simple closed curve

10.5.1 (Lem 65.1) Let G be a subspace of S^{2} that is a complete graph on four vertices a_{1}, …, a_{4}. Let C be the subgraph a_{1}a_{2}a_{3}a_{4}a_{1}, which is a simple closed curve. Let p and q be interior points of the edges a_{1}a_{3} and a_{2}a_{4}, respectively. Then

10.5.1.1 The points p and q lie in different components of S^{2} – C.

10.5.1.2 The inclusion j : C -> S^{2} – p – q induces an isomorphism of fundamental groups.

10.5.1.2.1 Pf) (Lem 64.1)(Lem 64.3)

10.5.1.3 Let C be a simple closed curve in S^{2}; let p and q lie in different components of S^{2} – C. Then the inclusion mapping j : C -> S^{2} – p – q induces an isomorphism of fundamental groups.

10.5.1.3.1 Pf) (Cor 58.5)

10.6 Cauchy integral formula

10.6.1 Winding number of f with respect to a ( a point a not in the image of a loop f in \mathbb{R}^{2}) : Let g(s) = [f(s) – a] / \Vert f(s) – a \Vert then g is a loop in S^{1}. Let p : \mathbb{R} -> S^{1} be the standard covering map, and let \bar{g} be a lifting of g to S^{1}. Because g is a loop, the difference \bar{g}(1) - \bar{g}(0) is an integer. This integer is winding number of f w.r.t. a .

10.6.1.1 n( f, a ) <- winding number of f with respect to a

10.6.2 Free homotopy F between the loops f_{0} and f_{1} (a continuous map F : I \times I -> X ) : Let F(0,t) = F(1,t) for all t. Then for each t, the map f_{t}(s) = F(s,t) is a loop in X.

10.6.2.1 It is a homotopy of loops in which the base point of the loop is allowed to move during the homotopy.

10.6.3 (Lem 66.1) Let f be a loop in \mathbb{R}^{2} – a.

10.6.3.1 If \bar{f} is the reverse of f, then n(\bar{f} , a) = -n (f, a).

10.6.3.2 If f is freely homotopic to f’, through loops lying in \mathbb{R}^{2}-a, then n(f,a) = n(f’, a).

10.6.3.3 If a and b lie in the same component of \mathbb{R}^{2} – f(1), then n(f,a) = n(f,b).

10.6.3.4 Simple loop f (a loop f in X) : f(s) = f(s’) only if s = s’ or if one of the points s, s’ is 0 and the other is 1.

10.6.3.4.1 If f is a simple loop, its image set is a simple closed curve in X.

10.6.3.5 (Thm 66.2) Let f be a simple loop in \mathbb{R}^{2}. If a lies in the unbounded component of \mathbb{R}^{2} – f(1), then n(f,a) = 0; while if a lies in the bounded component, n(f,a) = \mp 1.

10.6.3.5.1 Pf) (Thm 54.5)

10.6.3.6 Counterclockwise loop f (a simple loop f in \mathbb{R}^{2}) : n(f,a) = +1 for some a (and hence for every a) in the bounded componenet of \mathbb{R}^{2} – f(1).

10.6.3.6.1 Clockwise loop f if n(f,a) = -1.

10.6.3.7 (Lem 66.3) Let f be a piecewise-differentiable loop in the complex plane; let a be a point not in the image of f. Then n(f,a) = \frac{1}{2 \pi i} \int_{f} \frac{dz}{z-a}

10.6.3.7.1 Definition of the winding number of f : \frac{1}{2 \pi i} \int_{f} \frac{dz}{z-a}

10.6.3.8 (Cauchy integral formula-classical version) (Thm 66.4) Let C be a simple closed piecewise-differentiable curve in the complex plane. Let B be the bounded component of \mathbb{R}^{2} – C. If F(z) is analytic in an open set \Omega that contains B and C, then for each point a of B, F(a) = \mp \frac{1}{2 \pi i} \int_{C} \frac{F(z)}{z-a} dz

10.6.3.8.1 + sign if C is oriented counterclockwise, and – otherwise.

Seifert-van Kampten Theorem

11 Seifert-van Kampten Theorem

11.1 Direct sum of abelian groups

11.1.1 G_{\alpha} generates G ( an indexed family {G_{\alpha}}{\alpha \in J} of subgroups of an abelian group G) : Every element x of G can be written as a finite sum of elements of the groups G{\alpha}

11.1.1.1 G is the sum of the groups G_{\alpha}. G = \sum_{\alpha \in J} G_{\alpha}

11.1.1.2 Direct sum G of the groups G_{\alpha} : groups G_{\alpha} generate G, and for each x \in G the expression x = \sum x_{\alpha} for x is unique.

11.1.1.2.1 There is only one J-tuple (x_{\alpha})(\alpha \in J) with x{\alpha} = 0 for all but finitely many \alpha s.t. x = \sum x_{\alpha}.

11.1.1.3 G = \bigoplus_{\alpha \in J} G_{\alpha} or in the finite case, G = G_{1} \oplus \cdots \oplus G_{n}.

11.1.2 (Extension condition for direct sums) (Lem 67.1) Let G be an abelian group; let {G_{\alpha} } be a family of subgroups of G. If G is the direct sum of the groups G_{\alpha}, then G satisfies the following condition:

11.1.2.1 Given any abelian group H and any family of homomorphisms h_{\alpha} : G_{\alpha} -> H, there exists a homomorphism h : G -> H whose restriction to G-{\alpha} equals h_{\alpha}, for each \alpha.

11.1.2.1.1 h is unique.

11.1.2.2 Conversely, If the groups G_{\alpha} generate G and the extension condition holds, then G is the direct sum of the groups G_{\alpha}.

11.1.2.3 (Cor 67.2)Let G = G_{1} \oplus G_{2}. Suppose G_{1} is the direct sum of subgroups H_{\alpha} for \alpha \in J, and G_{2} is the direct sum of subgroups H_{\beta} for \beta \in K, where the index sets J and K are disjoint. Then G is the direct sum of the subgroups H_{\gamma}, for \gamma \in J \cup K.

11.1.2.4 (Cor 67.3) If G = G_{1} \oplus G__{2}, then G/G_{2} is isomorphic to G_{1}.

11.1.3 External direct sum G of the groups G_{\alpha} relative to the monomorphisms i_{\alpha}. ( an indexed family {G_{\alpha}}{\alpha \in J} of subgroups of an abelian group G) Let i{\alpha} : G_{\alpha} -> G is a family of monomorphisms, such that G is the direct sum of the groups i_{\alpha} (G_{\alpha}).

11.1.4 (Thm 67.4) Given a fimly of abelian groups {G_{\alpha}}{\alpha \in J}, there exists an abelian group G and a family of monomorphisms i{\alpha} : G_{\alpha} -> G s.t. G is the direct sum of the groups i_{\alpha} (G_{\alpha}).

11.1.5 (Lem 67.5) Let {G_{\alpha}}{\alpha \in J} be an indexed family of abelian groups; Let G be an abelian group; Let i{\alpha} : G_{\alpha} -> G be a family of homomorphisms. If each i_{\alpha} is a monomorphism and G is the direct sum of the groups i_{\alpha} (G_{\alpha})., then G satisfies the following extension condition:

11.1.5.1 Given any abelian group H and any family of homomorphisms h_{\alpha} : G_{\alpha} -> H, there exists a homomorphism h : G -> H whose restriction to G-{\alpha} equals h_{\alpha}, for each \alpha

11.1.5.1.1 h is unique.

11.1.5.2 Conversely, If the groups i_{\alpha} (G_{\alpha}) generate G and the extension condition holds, then each i_{\alpha} is a monomorphism, and G is the direct sum of the groups G_{\alpha}.

11.1.6 (Uniqueness of direct sums) (Thm 67.6) Let {G_{\alpha}}{\alpha \in J} be an indexed family of abelian groups; Let G and G’ be an abelian group; Let i{\alpha} : G_{\alpha} -> G and i’{\alpha} : G{\alpha} -> G’ be a family of monomorphisms, such that G is the direct sum of the groups i_{\alpha} (G_{\alpha}) and G’ is the direct sum of the groups i’{\alpha} (G{\alpha}).. Then there is a unique isomorphism \phi : G -> G’ s.t. \phi \bullet i_{\alpha} = i’_{\alpha} for each \alpha.

11.1.6.1 pf) (Lem 67.5)

11.1.7 Free abelian group G having the elements {a_{\alpha}} as a basis. ( an indexed family {a_{\alpha}} of elements of an abelian group G) : Let G_{\alpha} be the subgroup of G generated by a_{\alpha}. If the groups G_{\alpha} generates, we also say that elements a_{\alpha}} generate G. If each group G_{\alpha} is infinite cyclic, and if G is the direct sum of the groups G_{\alpha}.

11.1.8 (Lem 67.6) Let G be an abelian group; let {a_{\alpha}}{\alpha \in J} be a family of elements of G that generates G. Then G is a free abelian group with basis {a{\alpha}} iff for any abelian group H and any family {y_{\alpha}} of elements of H, there is a homomorphism h of G into H s.t. h(a_{\alpha}) = y_{\alpha} for each \alpha. In such case, h is unique.

11.1.8.1 pf) (Lem 67.1)

11.1.9 (Thm 67.8) If G is a free abelian group with basis {a_{1}, …., a_{n}}, then n is uniquely determined by G.

11.1.10 Rank of G ( a free abelian group with a finite basis) : the number of elements in a basis of G.

11.2 Free products of Groups

11.2.1 Let G be a group. Let {G_{\alpha}}{\alpha \in J} be a family of subgroups of G, and {G{\alpha}}_{\alpha \in J} generate G.

11.2.1.1 word of length n in the groups G_{\alpha} : a finite sequence (x_{1}, …, x_{n}) of elements of the groups G_{\alpha} s.t. x = x_{1} \cdots x_{n}.

11.2.1.1.1 Represents the element x of G.

11.2.1.2 Reduced word : a word representing x of the form (y_{1}, …, y_{m}) where no group G_{\alpha} contains both y_{i} and y_{i+1}, and where y_{i} \neq 1 for all i.

11.2.2 G is the free product of the groups G_{\alpha} ( an indexed family {G_{\alpha}}{\alpha \in J} of subgroups of an abelian group G that generates G) : Suppose that G{\alpha} \cap G_{\beta} consists of the identity element alone whenever \alpha \neq \beta. For each x \in G, there is only one reduced word in the groups G_{\alpha} that represents x.

11.2.2.1 G = \prod_{\alpha \in J}^{*} G_{\alpha} or in the finite case, G = G_{1} * \cdots * G_{n}.

11.2.3 (Lem 68.1) a family {G_{\alpha}} } of subgroups of a group G. If G is the free product of the groups G_{\alpha}, then G satisfies the following condition:

11.2.3.1 Given any group H and any family of homomorphisms h_{\alpha} : G_{\alpha} -> H, there exists a homomorphism h : G -> H whose restriction to G_{\alpha} equals h_{\alpha}, for each \alpha.

11.2.3.1.1 h is unique

11.2.4 External free product G of the groups G_{\alpha} relative to the monomorphisms i_{\alpha}. ( an indexed family {G_{\alpha}}{\alpha \in J} of groups, a group G) Let i{\alpha} : G_{\alpha} -> G is a family of monomorphisms, such that G is the free product of the groups i_{\alpha} (G_{\alpha}).

11.2.5 (Lem 68.2)Given a family {G_{\alpha}}{\alpha \in J} of groups, There exists a group G and a family of monomorphisms i{\alpha} : G_{\alpha} -> G s.t. G is the free product of the groups i_{alpha}(G_{\alpha}).

11.2.6 (Extension condition for ordinary free products) (Lem 68.3) Let {G_{\alpha}} be a family of groups; Let G be a group ; Let i_{\alpha} : G_{\alpha} -> G be a family of homomorphisms. If each i_{\alpha} is a monomorphism and G is the free product of the groups i_{\alpha}(G_{\alpha}, then G satisfies the following condition:

11.2.6.1 Given any group H and any family of homomorphisms h_{\alpha} : G_{\alpha} -> H, there exists a homomorphism h : G -> H whose restriction to G_{\alpha} equals h_{\alpha}, for each \alpha.

11.2.6.1.1 h is unique

11.2.7 (Uniqueness of free products)( Thm 68.4) Let {G_{\alpha}}{\alpha \in J} be a family of groups; Let G and G’ be groups; Let i{\alpha} : G_{\alpha} -> G and i’{\alpha} : G{\alpha} -> G’ be a family of monomorphisms, such that the families [ i_{\alpha} (G_{\alpha}) ] and [ i’{\alpha} (G{\alpha}) ] generate G and G’. If both G and G’ have the extension property stated in the preceding lemma, then there is a unique isomorphism \phi : G -> G’ s.t. \phi \bullet i_{\alpha} = i’_{\alpha} for all \alpha.

11.2.8 (Lem 68.5) Let {G_{\alpha}} be a family of groups; Let G be a group ; Let i_{\alpha} : G_{\alpha} -> G be a family of homomorphisms. If rhe extension condition of (Lem 68.3) holds, then each i_{\alpha} is a monomorphism and G is the free product of the groups i_{\alpha}(G_{\alpha}

11.2.8.1 (Cor 68.6) Let G = G_{1} * G_{2}. Suppose G_{1} is the free product of the subgroups H_{\alpha} for \alpha \in J, and G_{2} is the free product of subgroups H_{\beta} for \beta \in K. If the index sets J and K are disjoint, then G is the free product of the subgroups {H_{\gamma}}{\gamma \in J \cup K}.

11.2.9 (Thm 68.7) Let G = G_{1} * G_{2}. Let N_{i} be a normal subgroup of G_{i}, for i = 1,2. If N is the least normal subgroup of G that contains N_{1} and N_{2}, then G / N \cong (G_{1} / N_{1}) * (G_{2} / N_{2}).

11.2.9.1 (Cor 68.8) If N is the least normal subgroup of G_{1} * G_{2} that contains G_{1}, then (G_{1} * G_{2}) / N \cong G_{2}.

11.2.10 (Lem 68.9) Let S be a subset of the group G. If N is the least normal subgroup of G containing S, then N is generatied by all conjugates of elements of S.

11.3 Free groups

11.3.1 Let G be a group; let {a_{\alpha}} be a family of elements of G, for \alpha \in J. We say trhe elements {a_{\alpha}} generate G if every element of G can be written a s a product of powers of the elements a_{\alpha}.

11.3.1.1 If the family {a_{\alpha}} is finite, we say G is finitely generated.

11.3.2 Let {a_{\alpha}} be a family of elements of a group G. Suppose each a_{\alpha} generates an infinite cyclic subgroup G_{\alpha} of G. If G is a free product of the groups {G_{\alpha}}, then G is said to be a free group.

11.3.2.1 the family {a_{\alpha}} is called system of free generators for G.

11.3.3 (Lem 69.1) Let G be a group; let {a_{\alpha}}{\alpha \in J} be a family of elements of G. If G is a free group with system of free generators {a{\alpha}}, then G satisfies the following condition(*) : Furthermore, h is unique.

11.3.3.1 (*) Given any group H and any family {y_{\alpha}} of elements of H, there exists a homomorphism h : G -> H whose h(a_{\alpha}) = y_{\alpha}, for each \alpha.

11.3.3.2 If the extension condition (*) holds, then G is a free group with system of free generators {a_{\alpha}}.

11.3.4 (Thm 69.2) Let G = G_{1} * G_{2}, where G_{1} and G_{2} are free groups with {a_{\alpha}}{\alpha \in J} and {a{\alpha}}{\alpha \in K} as respective systems of free generators. If J and K are disjoint, then G is a free group with {a{\alpha}}_{\alpha \in J \cup K} as a system of free generators.

11.3.5 Let {a_{\alpha}}{\alpha \in J} be an arbitrary indexed family. Let G{\alpha} denote the set of all symbols of the form a_{\alpha}^{n} for n \in \mathbb{Z}. We make G_{\alpha} into a group by defining a_{\alpha}^{n} \cdot a_{\alpha}^{n} = a_{\alpha}^{n+m}. Then a_{\alpha}^{0} is the identity element of G_{\alpha}, and a_{\alpha}^{-n} is the inverse of a_{\alpha}^{n} The external free product of the groups {G_{\alpha}} is called the free group on the elements a_{\alpha}.

11.3.5.1 We denote a_{\alpha}^{1} simply by a_{\alpha}.

11.3.6 Let G be a group. If x , y \in G, we denote by [x,y] the element [x,y] = xyx^{-1}y^{-1} of G; it is called the commutator of x and y. The subgroup of G generated by the set of all commutators in G is called the commutator subgroup of G and denoted [G,G].

11.3.7 (Lemma 69.3) Given G, the subgroup [G,G] is a normal subgroup of G and the quotient group G/[G,G] is abelian, If h:G->H is any homomorphism from G to an abelian group H, then the kernel of h contained [G,G], so h induces a homomorphism k : G/[G,G] -> H.

11.3.8 (Thm 69.4) If G is a free group with free generators a_{\alpha}, then G/[G,G] is a free abelian group with basis [a_{\alpha}], where [a_{\alpha]} denotes the coset of a_{\alpha} in G/[G,G].

11.3.8.1 (Cor 69.5) If G is a free group with n free generators, then any system of free generators for G has n elements.

11.3.9 If G is a group, a presentation of G consists of a family {a_{\alpha}} of generators for G, along with a complete set {r_{\beta}} of relations for G, where each r_{\beta is an element of the free group on the set {a_{\alpha}}.

11.3.9.1 If the family {a_{\alpha}} is finite, the G is finitely generated. If both the families {a_{\alpha} } and {r_{\beta}} are finite, then G is said to be finitely presented, and these families form a finite presentation for G.

11.4 Seifert-van Kampen Theorem

11.4.1 (Seifert-van Kampen Theorem) (Thm 70.1) : Let X = U \cup V, where U and V are open in X; assume U, V, and U \cap V are path connected; let x_{0} \in U \cap V. Let H be a group, and let \phi_{1} : \pi_{1}(U,x_{0}) -> H and \phi_{2} : \pi_{1}(V,x_{0}) -> H be homomorphisms. Let i_{1}, i_{2}, j_{1}, j_{2} be the homomorphisms indicated in the following diagram, each induced by inclusion. If \phi_{1} \bullet i_{1} = \phi_{2} \bullet i_{2}, then there is a unique homomorphism \Phi : \phi_{1} : \pi_{1}(X,x_{0}) -> H s.t. \Phi \bullet j_{1} = \phi_{1} and \Phi \bullet j_{2} = \phi_{2}.

11.4.1.1 \begin {tikzpicture} \matrix (m) [matrix of math nodes,row sep=3em,column sep=4em,minimum width=2em] {& \pi_{1} (U,x_{0}) &\ \pi_{1} (U,x_{0})&\pi_{1}(U,x_{0}) & H \& \pi_{1} (U,x_{0})&\}; \path[-stealth] (m-1-2) edge node [right] {$ j_{1} $} (m-2-2) edge node[above] {$ \phi_{1} $} (m-2-3) (m-2-1) edge node [above] {$ i_{1} $} (m-1-2) edge (m-2-2) edge node [below] {$ i_{2} $} (m-3-2) (m-2-2) edge node [above] {$ \Phi $} (m-2-3) (m-3-2) edge node [right] {$ j_{2} $} (m-2-2) edge node[below] {$ \phi_{2} $} (m-2-3); \end {tikzpicture}

11.4.2 (Seifert-van Kampen theorem, classical version) (Thm 70.2) Assume the hypotheses of the preceding theorem. Let j : \pi_{1} (U,x_{0}) * \pi_{1}(V,x_{0}) -> \pi_{1} (X,x_{0}) be the homomorphism of the free product that extends the homomorphisms j_{1} and j_{2} induced by inclusion. Then j is surjective, and its kernel is the least normal subgroup N of the free product that contains all elements represented by words of the form (l_{1}(g)^{-1} i_{2}(g)), for g \in \pi_{1} (U \cap V ,x_{0}).

11.4.2.1 (Cor 70.3) Assume the hypotheses of the Seifert-van Kampen theorem. If U \cap V is simply connected, then there is an isomorphism k : pi (U,x_{0}) * \pi_{1}(V,x_{0}) -> \pi_{1} (X,x_{0})

11.4.2.2 (Cor 70.4) Assume the hypotheses of the Seifert-van Kampen theorem. If V is simply connected, then there is an isomorphism k : \pi_{1} (U,x_{0})/N -> \pi_{1} (X,x_{0}) where N is the least normal subgroup of \pi_{1} (U,x_{0}) containing the image of homomorphism i_{1} : \pi_{1} (U \cap V ,x_{0}) -> \pi_{1} (U,x_{0})

11.5 Fundamental group of a wedge of circles

11.5.1 Wedge of the circles S_{1} , …, S_{n} (a Hausdorff space X ) : X is the union of the subspaces S_{1} , …, S_{n} each of which is homeomorphic to the unit circle S^{1}. There is a point of X s.t. S_{i} \cap S_{j} = {p} whenever i \neq j.

11.5.2 (Thm 71.1) Let X be the wedge of the circles S_{1} , …, S_{n}; let p be the common point of these circles. then \pi_{1} (X,p) is a free group. If f_{i} is a loop in S_{i} that represents a generator of \pi_{1} (S_{i},p), then the loops f_{1} , …, f_{n} represent a system of free generators for \pi_{1} (X,p).

11.5.3 \Tau Coherent with X_{\alpha} ( a topology of a space X which is union of the subspaces X_{\alpha} for \alpha \in J) : a subset C of X is closed in X if C \cap X_{\alpha} is closed in X_{\alpha} for each \alpha.

11.5.3.1 equivalent condition is that a set be open in X if its intersection with each X_{\alpha} is open in X_{\alpha}.

11.5.4 wedge X of the circles S_{\alpha} ( a space X that is the union of the subspaces X_{\alpha} for \alpha \in J) : Let each of S_{\alpha} be homeomorphic to the unit circle. Assume there is a point p of X s.t. S_{\alpha} \cap S_{\beta} = {p} whenever \alpha \neq \beta. If the topology of X is coherent with the subspaces S_{\alpha}, then X is called the wedge of the circles S_{\alpha}.

11.5.5 (Lem 71.2) Let X be the wedge of the circles S_{\alpha}, for \alpha \in J. The X is normal.

11.5.5.1 Any compact subspace of X is contained in the union of finitely many circles S_{\alpha}.

11.5.6 (Thm 71.3) Let X be the wedge of the circles S_{\alpha}, for \alpha \in J; Let p be the common point of these circles. Then \pi_{1} (X,p) is a free group. If f_{\alpha} is a loop in S_{\alpha} representing a generator of \pi_{a} (S_{\alpha} , p), then the loops {f_{\alpha}} represent a system of free generators for \pi_{1} (X,p).

11.5.7 (Lem 71.4) Given an index set J, there exists a space X that is a wedge of circles S_{\alpha} for \alpha \in J.

11.6 Adjoining a Two-cell

11.6.1 (Thm 72.1)Let X be a Hausdorff space; let A be a closed path-connected subspace of X. Suppose that there is a continuous map h : B^{2} -> X that maps Int B^{2} bijectively onto X-A and maps S^{1} = Bd B^{2} into A. Let p \in S^{1} and let a = h(p); let k : (S^{1}, p) -> (A,a) be the map obtained by restricting h. Then the homomorphism i_{} : \pi_{1} (A,a) -> \pi_{1} (X,a) induced by inclusion is surjective, and its kernel is the least normal subgroup of \pi_{1} (A,a) containing the image of k_{} : \pi_{1} (S^{1},p) -> \pi_{1} (A,a)

11.6.1.1 The fundamental group of X is obtained from the fundamental group of A by killing off the class k_{*} [ f ] , where [f] generates \pi_{1} (S^{1},p).

11.7 Fundamental groups of the Torus and the Dunce cap

11.7.1 (Thm 73.1) The fundamental group of the torus has a presentation consisting of two generators \alpha, \beta and a single relation \alpha \beta \alpha^{-1} \beta^{-1} .

11.7.1.1 (Cor) The fundamental group of the torus is a free abelian group of rank 2.

11.7.2 n-fold dunce cap X : Let n be a positive integet with n > 1. Let r : S^{1} -> S^{1} be rotation through the angle \frac{2\pi}{n}, mapping the point (cos\theta, sin\theta) to the point (cos(\theta +\frac{2\pi}{n} ) , sin(\theta+\frac{2\pi}{n})). Form a quotient space X from the unit ball B^{2} by identifying each point x of S^{1} with the points r(x) , r^{2}(x), …, r^{n-1}(x).

11.7.2.1 X is a compact Hausdorff space.

11.7.3 (Lem 73.3) Let \pi : E -> X be a closed quotient map. If E is normal, then so is X.

11.7.4 (Thm 73.4) The fundamental group of the n-fold dunce cap is a cyclic group of order n.

Classification of Surfaces

12 Classification of Surfaces

12.1 Fundamental Groups of surfaces

12.1.1 Polygonal region P determined by the points p_{i} : Given a point c of \mathbb{R}^{2} and given a >0, consider the circle of radius a in \mathbb{R}^{2} with center at c. Given a finite sequence \theta_{0} < \theta_{1} < \cdots < \theta_{n} of real numbers, where n \ge 3 and \theta_{n} = \theta_{0} + 2\pi, consider the points p_{i} = c + a(cos\theta_{i}, sin\theta_{i}), which lie on this circle. They are numbered in counterclockwise order around circle, and p_{n} = p_{0}. The line through p_{i-1} and p_{i} splits the plane into two closed half-planes; let H_{i} be the one that contains all the points p_{k}. Then P is P = H_{1} \cap \cdots \cap H_{n}.

12.1.1.1 vertices of P : points p_{i}

12.1.1.2 edge of P : the line segment joining p_{i-1} and p_{i}.

12.1.1.3 Bd P : Union of the edges of P

12.1.1.4 Int P : P – Bd P

12.1.1.5 If p is any point of Int P, then P is the union of all line segments joining p and points of Bd P, and that two such line segments intersects only in the points p.

12.1.2 Given a line segment L in \mathbb{R}^{2}

12.1.2.1 orientation of L is simply an ordering of its end points

12.1.2.1.1 the first, a, is the initial point of the oriented line segments.

12.1.2.1.2 the second, b, is the final point of the oriented line segments.

12.1.2.1.3 L is oriented from a to b

12.1.2.1.4 If L’ is another line segment oriented from c to d, then the positive linear map of L onto L’ is the homeomorphism h : L -> L’; x = (1-s)a + sb -> h(x)0 = (1-s)c + sd

12.1.3 Labeling of the edges of P ( a polygonal region P in the plane) : a map from the set of edges of P to a set of labels S. given an orientation of each edge of P , and given a labeling of the edges of P, we define an equivalence relation on the points of P as follows: Each point of Int P is equivalent only to itself. Given any two edges of P that have the same label, let h be the positive linear map of one onto the other, and define each point x of the first edge to be equivalent to the point h(x) of the second edge. This relation generates an equivalence relation on P. The quotient space X obtained from this equivalence relation is said to have been obtained by pasting the edges of P together according to the given orientations and labeling.

12.1.4 Let P be a polygonal region with successive vertices p_{0}, .., p_{n}, where p_{0} = p_{n}. Given orientations and a labeling of the edges of P, let a_{1}, .., a_{m} be the distinct labels that are assigned to the edges of P. For each k, let a_{ik} be the label assigned to the edge p_{k-1}p_{k}, and let \epsilon_{k} = +1 or -1 according as the orientation assigned to this edge goes from p_{k-1} to p_{k} or the reverse. Then the number of edges of P, the orientations of the edges, and the labeling are completely specified by the symbol w = (a_{i_{1}})^{\epsilon_{1}}(a_{i_{2}})^{\epsilon_{2}} \cdots (a_{i_{n}})^{\epsilon_{n}}. We call this symbol a labeling scheme of length n for the edges of P; it is simply a sequence of labels with exponents +1 or -1.

12.1.5 (Thm 74.1) Let X be a space obtained from a finite collection of polygonal regions by pasting edges together according to some labelling scheme, Then X is a compact Hausdorff space.

12.1.6 (Thm 74.2) Let P be a polygonal region; let w = (a_{i_{1}})^{\epsilon_{1}} (a_{i_{2}})^{\epsilon_{2}} \cdots (a_{i_{n}})^{\epsilon_{n}} be a labeling scheme for the edges of P. Let X be the resulting quotient space; let \pi : P ->X be the quotient map. If \pi maps all the vertices of P to a single point x_{0} of X, and if a_{1}, …, a_{k} are the distinct labels that appear in the labeling scheme, then \pi_{1}(X,x_{0}) is isomorphic to the quotient of the free groups on k generators \alpha_{1} .. \alpha_{k} by the least normal subgroup containing the element (\alpha_{i_{1}})^{\epsilon_{1}} (\alpha_{i_{2}})^{\epsilon_{2}} \cdots (\alpha_{i_{n}})^{\epsilon_{n}}

12.1.7 Consider the space obtained from a 4n-sided polygonal region P by means of the labeling scheme (a_{1}b_{1}a_{1}^{-1}b_{1}^{-1})(a_{2}b_{2}a_{2}^{-1}b_{2}^{-1}) \cdots (a_{n}b_{n}a_{n}^{-1}b_{n}^{-1}). This space is called the n-fold connected sum of tori, or simply the n-fold torus, and denoted T# \cdots #T.

12.1.8 (Thm 74.3) Let X denote the n-fold torus. Then \pi_{1}(X,x_{0}) is isomorphic to the quotient of the free group on the 2n generators \alpha_{1}, \beta{1}, …, \alpha_{n}, \beta{n} by the least normal subgroup containing the element [\alpha_{1}, \beta{1}] [\alpha_{2}, \beta{2}] \cdots [\alpha_{n}, \beta{n}] where [\alpha,\beta] = \alpha \beta \alpha^{-1}\beta^{-1}.

12.1.9 Let m>1. Consider the space obtained from a 2m-sided polygonal region P in the plane by means of the labeling scheme (a_{1}a_{1})(a_{2}a_{2})\cdots(a_{m}a_{m}). This space is called the m-fold connected sum of the projective planes, or simply the m-fold projective plane, and denoted P^{2}# \cdots #P^{2}.

12.1.10 (Thm 74.4) Let X denote the m-fold projective plane. Then \pi_{1}(X,x_{0}) is isomorphic to the quotient of the free group on m generators \alpha_{1}, … ,\alpha_{m} by the least normal subgroup containing the element (a_{1})^{2} (a_{2})^{2}\cdots(a_{m})^{2}.

12.2 Homology of Surfaces

12.2.1 If X is a path-connected space, Let H_{1}(X) = pi_{1}(X, x_{0}) / [\pi_{1}(X,x_{0}), \pi_{1}(X,x_{0}) ]. We call H_{1} (X) the first homology group of X.

12.2.1.1 we omit the basepoint from the notation because there is a unique path-induced isomorphism between the abelianized fundamental groups based at two different points.

12.2.2 Homology groups of X H_{n}(X)

12.2.3 (Thm 75.1) Let F be a group; let N be a normal subgroup of F; let q : F -> F/N be the projection. The projection homomorphism p : F -> F / [F,F] induces an isomorphism \phi : q(F) / [q(F), q(F)] -> p(F)/p(N).

12.2.3.1 (Cor 75.2) Let F be a free group with free generators \alpha_{1}, …, \alpha_{n}; let N be the least normal subgroup of F containing the elements x of F; let G = F/N. Let p : F -> F/[F,F] be projection. Then G/[G,G] is isomorphic to the quotient of F/[F,F], which is free abelian with basis p(\alpha_{1}), …, p(\alpha_{n}), by the subgroup generated by p(x).

12.2.4 (Thm 75.3) If X is the n-fold connected sum of tori, then H_{1}(X) is a free abelian group of rank 2n.

12.2.5 (Thm 75.4) If X is the m-fold connected sum of projective planes, then the torsion subgroup T(X) of H_{1}(X) has order 2, and H_{1}(X)/T(X) is a free abelian group of rank m-1.

12.2.6 (Thm 75.5) Let T_{n} and P_{m} denote the n-fold connected sum of tori and the m-fold connected sum of projective planes, respectively. Then the surfaces S^{2}; T_{1}, T_{2}, …; P_{1}, P_{2}, … are topologically distinct.

12.3 Cutting and Pasting

12.3.1 (Theorem 76.1) Suppose X is the space obtained by pasting the edges of m polygonal regions together according to the labeling scheme y() {0}y{1}, w_{2}, .., w_{m}. Let c be a label not appearing in this scheme. If both y_{0} and y_{1} have length at least two, then X can also be obtained by pasting the edges of m+1 polygonal regions together according to the scheme () y_{0}c^{-1} , cy_{1}, w_{2}, …, w_{m}. Conversely if X is the space obtained from m+1 polygonal regions by means of the scheme (), it can also be obtained from m polygonal regions by means of the scheme() providing that c does not appear in scheme (*).

12.3.2 Elementary operations on schemes

12.3.2.1 On a labeling scheme w_{1}, …, w_{m} without affecting the resulting quotient space X.

12.3.2.1.1 Cut. One can replace the scheme w_{1} = y_{0}y_{1} by the scheme y_{0}c^{-1} and cy_{1}, provided c does not appear elsewhere in the total scheme and y_{0} and y_{1} have length at least two.

12.3.2.1.2 Paste. One can replace the scheme y_{0}c^{-1} and cy_{1} by the scheme y_{0}y_{1}, provided c does not appear elsewhere in the total scheme.

12.3.2.1.3 Relabel. One can replace all occurrences of any given label by some other label that does not appear elsewhere in the scheme. Similarly, one can change the sign of the exponent of all occurrences of a given label a; this amounts to reversing the orientations of all the edges labeled “a”. Neither of these alterations affect the pasting map.

12.3.2.1.4 Permute : One can replace any one of the schemes w_{i} by a cyclic permutation of w_{i}. Specifically, if w_{i} = y_{0}y_{1}, we can replace w_{i} by y_{1}y_{0}. This amount to renumbering the vertices of the polygonal region P_{i} so as to begin with a different vertex; it does not affect the resulting quotient space.

12.3.2.1.5 Flip. One can replace any one of the schemes w_{i} = (a_{i_{1}}) ^{\epsilon_{1}} (a_{i_{2}})^{\epsilon_{2}} \cdots (a_{i_{n}})^{\epsilon_{n}} by its formal inverse w^{-1}{i} = (a{i_{n}})^{-\epsilon_{n \cdots (a_{i_{1}})^{-\epsilon_{1}} This amounts to simply to “flipping the polygonal region P_{i} over”. The order of vertices is reversed, and so is the orientation of each edge. The quotient space X is not affected.

12.3.2.1.6 Cancel. One can replace the scheme w_{i} = y_{0}aa^{-1}y_{1} by the scheme y_{0}y_{1}, provided a does not appear elsewhere in the total scheme and both y_{0} and y_{1} have length at least two.

12.3.2.1.7 Uncancel. This is the reverse of Cancel. It replaces the scheme y_{0}y_{1} by the scheme y_{0}aa^{-1}y_{1}, where a is a label that does not appear elsewhere in the total scheme.

12.3.3 Define two labeling schemes for collections of polygonal regions to be equivalent if one can be obtained from the other by a sequence of elementary scheme operations.

12.3.3.1 Since each elementary operation has as its inverse another such operation, this is an equivalence relation.

12.4 Classification Theorem

12.4.1 Suppose w_{1}, …, w_{k} is a labeling scheme for the polygonal regions P_{1}, …, P_{k} If each label appears exactly twice in this scheme, we call it a proper labeling scheme. If one applies any elementary operation to a proper scheme, one obtains another proper scheme.

12.4.2 Let w be a proper labeling scheme for a single polygonal region, we say w is of torus type if each label in it appears once with exponent +1 and once with exponent -1. Otherwise, we say w is of projective type.

12.4.3 (Lem 77.1) Let w be a proper scheme of the form w = [y_{0}]a[y_{1}]a[y_{2}] where some of the y_{i} may be empty. Then one has the equivalence w ~ aa[y_{0}y_[1]^{-1}y_{2}] where y_{1}^{-1} denotes the formal inverse of y_{1}.

12.4.3.1 (Cor 77.2) If w is a scheme of projective type, then w is equivalent to a scheme of the same length having the form (a_{1}a_{1})(a_{2}a_{2}) \cdots (a_{k}a_{k})w where k \ge 1 and w_{1} is either empty or of torus type.

12.4.4 (Lem 77.3) Let w be a proper scheme of the form w = w_{0}w_{1}, where w_{1} is a scheme of torus type that does not contain two adjacent terms having the same label. Then w is equivalent to a scheme of the form w_{0}w_{2}, where w_{2} has the same length as w_{1} and has the form w_{2} = aba^{-1}b^{-1} w_{3} where w_{3} is of torus type or is empty.

12.4.5 (Lem 77.4) Let w be a proper scheme of the form w = w_{0}(cc)(aba^{-1}b^{-1})w_{1}. Then w is equivalent to the scheme w’ = w_{0}(aabbcc)w_{1}.

12.4.6 (Classification theorem)(Thm 77.5) Let X be the quotient space obtained from a polygonal region in the plain by pasting its edges together in pairs. Then X is homeomorphic either to S^{2}, to the n-fold torus T_{n}, or to the m-fold projective plane P_{m}.

12.5 Constructing Compact surfaces

12.5.1 Let X be a compact Hausdorff space. A curved triangle in X is a subspace A of X and a homeomorphism h : T -> A, where T is a closed triangular region in the plane. If e is an edge of T, then h(e) is said to be an edge of A; if v is a vertex of T, then h(v) is said to be a vertex of A. A triangulation of X is a collection of curved triangles A_{1}, …, A_{n} in X whose union is X s.t. for i \neq j, the intersection A_{i} \cap A_{j} is either empty, or a vertex of both A_{i} and A_{j}, or an edge of both. Furthermore, if h_{i} : T_{i} -> A_{i} is the homeomorphism associated with A_{i}, we require that when A_{i} \cap A_{j} is an edge e of both, then the map h_{j}^{-1}h_{i} defines a linear homeomorphism of the edge h_{i}^{-1}€ of T_{i} with the edge h_{j}^{-1}(e) of T_{j}. If X has a triangulation, It is said to be triangulable.

12.5.1.1 Every compact surface is triangulable.

12.5.2 (Thm 78.1) If X is a compact triangulable surface, then X is homeomorphic to the quotient space obtained from a collection of disjoint triangular regions in the plane by pasting their edges together in pairs.

12.5.3 (Thm 78.2) If X is a compact connected triangulable surface, then X is homeomorphic to a space obtained from a polygonal region in the plane by pasting the edges together in pairs.

Classification of Covering spaces

13 Classification of Covering spaces

13.1 Convention

13.1.1 The statement that p : E->B is a covering map will include the assumption that E and B are locally path connected and path connected.

13.1.2 We assume the general lifting correspondence theorem (Thm 54.6).

13.2 Equivalence of Covering spaces

13.2.1 Let p : E->B and p’ : E’->B be covering maps. They are said to be equivalent if there exists a homeomorphism h : E-> E’ s.t. p = p’ \bullet h. The homeomorphism h is called an equivalence of covering maps or an equivalence of covering spaces.

13.2.2 (General Lifting lemma) (Lem 79.1) Let p : E->B be a covering map; let p(e_{0}) = b_{0}. Let f : Y->B be a continuous map, with f(y_{0}) = b_{0}. Suppose Y is path connected and locally path connected. The map f can be lifted to a map \bar{f} : Y -> E s.t. \bar{f}(y_{0}) = e_{0} iff f_{}(pi_{1}(Y,y_{0})) \subset p_{}(pi_{1}(E,e_{0})). Furthermore, if such a lifting exists, it is unique.

13.2.3 (Thm 79.2) Let p : E->B and p’ : E’->B be covering maps; let p(e_{0}) = p’(e’{0}) = b{0}. There is an equivalence h : E->E’ s.t. h(e_{0}) = e’{0} iff the groups H{0} = p_{}(pi_{1}(E,e_{0})) and H’{0} = p’{}(pi_{1}(E’,e’_{0})) are equal. If h exists, it is unique.

13.2.4 If H_{1} and H_{2} are subgroups of a group G, they are said to be conjugate subgroups if H_{2} = \alpha \cdot H_{1} \cdot \alpha^{-1} for some element \alpha of G.

13.2.4.1 Conjugacy is an equivalence relation on the collection of subgroups of G.

13.2.4.2 Equivalence class of the subgroup H is called the conjugacy class of H.

13.2.5 (Lem 79.3) Let p:E->B be a covering map. Let e_{0} and e_{1} be points of p^{-1}(b_{0}) and let H_{i} = p_{*}(pi_{1}(E,e_{i})).

13.2.5.1 If \gamma is a path in E from e_{0} to e_{1}, and \alpha is the loop p \bullet \gamma in B, then the equation [\alpha] * H_{1} * [\alpha]^{-1} = H_{0} holds ; hence H_{0} and H_{1} are conjugate.

13.2.5.2 Conversely, given e_{0}, and given a subgroup H of \pi_{1}(B,b_{0}) conjugate to H_{0}, there exists a point e_{1} of p^{-1}(b_{0}) s.t. H_{1} = H.

13.2.6 (Thm 79.4) Let p:E->B and p’ : E’ -> B be covering maps; let p(e_{0}) = p’(e’{0}) = b{0}. The covering maps p and p’ are equivalent iff the subgroups H_{0} = p_{}(pi_{1}(E,e_{0})) and H’{0} = p’{}(pi_{1}(E’,e’{0})) of \pi{1} (B, b_{0}) are conjugate.

13.3 Universal covering space

13.3.1 Suppose p : E->B is a covering map, with p(e_{0}) = b_{0}. If E is simply connected, then E is called a universal covering space of B.

13.3.2 (Lem 80.1) Let B be path connected and locally path connected. Let p : E-> B be a covering map in the former sense (so that E is not required to be path connected). If E_{0} is a path component of E, then the map p_{0} : E_{0} -> B obtained by restricting p is a covering map.

13.3.3 (Lem 80.2) Let p, q and r be continuous maps with p = r \bullet q, as in the following diagram :

13.3.3.1 \begin{tikzpicture} \matrix (m) [matrix of math nodes,row sep=3em,column sep=4em,minimum width=2em] { X & Y \ Z & \}; \path[-stealth] (m-1-1)edge node [above] {$q$} (m-1-2) edge node [left] {$p$} (m-2-1) (m-1-2) edge node [below] {$r$} (m-2-1);\end{tikzpicture}

13.3.3.2 If p and r are covering maps, so is q.

13.3.3.3 If p and q are covering maps, so is r.

13.3.4 (Thm 80.3) Let p : E->B be a covering map, with E simply connected. Given any covering map r : Y->B, there is a covering map q : E->Y s.t. r \bullet q = p.

13.3.4.1 \begin{tikzpicture} \matrix (m) [matrix of math nodes,row sep=3em,column sep=4em,minimum width=2em] { E & Y \ B & \}; \path[-stealth] (m-1-1)edge node [above] {$q$} (m-1-2) edge node [left] {$p$} (m-2-1) (m-1-2) edge node [below] {$r$} (m-2-1);\end{tikzpicture}

13.3.4.2 This theorem shows why E is called a universal covering space of B; it covers every other covering space of B.

13.3.5 (Lem 80.4) Let p : E->B be a covering map; let p(e_{0}) = b_{0}. If E is simply connected, then b_{0} has a neighborhood U s.t. inclusion i : U ->B induces the trivial homomorphism i_{*} : \pi_{1}(U,b_{0}) -> \pi_{1}(B,b_{0}).

13.4 Covering Transformations

13.4.1 Given a covering map p : E->B, Consider the set of all equivalences of this covering space with itself; Such an equivalence is called a covering transformation.

13.4.1.1 Composites and inverses of covering transformations are covering transformations; so it forms a group of covering transformations and denoted \mathcal{C}(E,p, B).

13.4.2 Assume p : E->B is a covering map with p(e_{0}) = b_{0}; and Let H_{0} = p_{*}(\pi_{1}(E,e_{0})) throughout this section.

13.4.3 If H is a subgroup of a group G, then the normalizer of H in G is the subset of G defined by the equation N(H) = {g | gHg^{-1} = H}.

13.4.3.1 N(H) is a subgroup of G.

13.4.4 Given p : E->B with p(e_{0}) = b_{0}, let F be the set F = p^{-1}(e_{0}). Let \Phi : \ pi_{1}(B,b_{0})/H_{0} -> F be the lifting correspondence of (Thm 54.6); it is a bijection. Define also a correspondence \Psi : \mathcal{C} (E,p,B) ->F by setting \Psi (h) = h(e_{0}) for each covering transformation h : E->E.

13.4.4.1 since h is uniquely determined once its value at e_{0} is known, the correspondence \Psi is injective.

13.4.5 (Lem 81.1) The image of the map \Psi equals the image under \Phi of the subgroup N(H_{0}) / H_{0} of \pi_{1}(B,b_{0}) / H_{0}.

13.4.6 (Thm 81.2) The bijection \Phi ^{-1} \bullet \Psi : \mathcal{C} (E,p,B) -> N(H_{0})/H_{0} is an isomorphism of groups.

13.4.6.1 (Cor 81.3) The group H_{0} is a normal subgroup of \pi_{1}(B,b_{0}) iff for every pair of points e_{1} and e_{2} of p^{-1}(b_{0}), there is a covering transformation h : E->E with h(e_{1}) = e_{2}. In this case, there is an isomorphism \Phi ^{-1} \bullet \Psi : \mathcal{C} (E,p,B) -> \pi_{1}(B,b_{0}) /H_{0}.

13.4.6.2 (Cor 81.4) Let p : E->B be a covering map. If E is simply connected, then \mathcal{C} (E,p,B) \cong \pi_{1}(B,b_{0}).

13.4.7 If H_{0} is a normal subgroup of \pi_{1}(B,b_{0}), then p : E->B is called a regular covering map. (not related to ‘regular’)

13.4.8 Let X be a space, and let G be a subgroup of the group of homeomorphisms of X with itself. The orbit space X/G is defined to be the quotient space obtained from X by means of the equivalence relation x ~ g(x) for all x \in X and all g \in G. The equivalence class of x is called the orbit of x.

13.4.9 If G is a group of homeomorphisms of X, the action of G on X is said to be properly discontinuous if for every x \in X there is a neighborhood U o fx s.t. g(U) is disjoint from U whenever g \neq e. (Here e is the identity element of G) It follows that g_{0}(U) and g_{1}(U) are disjoint whenever g_{0} \neq g_{1}, for otherwise U and g^{-1}{0} g{1}(U) would not be disjoint.

13.4.10 (Thm 81.5) If p : X->B is a regular covering map and G is its group of covering transformations, then there is a homeomorphism k : X/G -> B s.t. p = k \bullet \pi, where \pi : X->X/G is the projection.

13.4.10.1 \begin{tikzpicture} \matrix (m) [matrix of math nodes,row sep=3em,column sep=4em,minimum width=2em] { X & \ X/G & B \}; \path[-stealth] (m-1-1)edge node [above] {$p$} (m-2-2) edge node [left] {$\pi$} (m-2-1) (m-2-1) edge node [below] {$k$} (m-2-2);\end{tikzpicture}

13.5 Existence of Covering spaces

13.5.1 A space B is said to be semilocally simply connected if for each b \in B, there is a neighborhood U of b s.t. the homomorphism i_{*} : \pi_{1}(U,b) -> \pi_{1}(B,b) induced by inclusion is trivial.

13.5.2 (Thm 82.1) Let B be path connected, locally path connected, and semilocally simply connected. Let b_{0} \in B. Given a subgroup H of \pi_{1}(B,b_{0}), there exists a covering map p : E->B and a point e_{0} \in p^{-1}(b_{0}) s.t. p_{*}(\pi_{1}(E,e_{0})) = H.

13.5.2.1 (Cor 82.2) The space B has a universal covering space iff B is path connected, locally path connected, and semilocally simply connected.

Applications to Group Theory

14 Applications to Group Theory

14.1 Covering Spaces of a Graph

14.1.1 A linear graph is a space X that is written as the union of a collection of subspaces A_{\alpha}, each of which is an arc, s.t.

14.1.1.1 The intersection A_{\alpha} \cup A_{\beta} of two arcs is either empty or consists of a single point that is an end point of each.

14.1.1.2 The topology of X is coherent with the subspaces A_{\alpha}.

14.1.1.2.1 The arcs A_{\alpha} are called the edges of X, and their interiors are called the open edges of X, their end points are called the vertices of X; we denote the set of vertices of X by X^{0}.

14.1.2 (Lem 83.1) Every linear graph X is Hausdorff; in fact it is normal.

14.1.3 Let X be a linear graph. Let Y be a subspace of X that is a union of edges of X. Then Y is closed in X and is itself a linear graph ; we call it a subgraph of X.

14.1.4 (Lem 83.2) Let X be a linear graph. If C is a compact subspace of X, there exists a finite subgraph Y of X that contains C. IF C is connected, Y can be chosen to be connected.

14.1.5 (Lem 83.3) If X is a linear graph, then X is locally path connected and semilocally simply connected.

14.1.6 (Thm 83.4) Let p : E->X be a covering map, where X i s a linear graph. If A_{\alpha} is an edge of X and B is a path component of p^{-1}(A_{\alpha}), then p maps B homeomorphically onto A_{\alpha}. Furthermore, the space E is a linear graph, with the path componenets of the spaces p^{-1}(A_{\alpha}) as its edges.

14.2 The fundamental group of a graph

14.2.1 An oriented edge e of a graph X is an edge of X together with an ordering of its vertices; the first is called the initial vertex, and the second, the final vertex, of e,. an edge path in X is a sequence e_{1}, …, e_{n} of oriented edges of X s.t. the final vertex of e_{i} equals the initial vertex of e_{i+1}, for i = 1, …, n-1. Such an edge path is entirely specified by the sequence of vertices x_{0}, .., x_{n}, where x_{0} is the initial vertex of e_{1} and x_{i} is the final vertex of e_{i} for i = 1,…,n. It is said to be an edge path from x_{0} to x_{n}. It is called a closed edge path if x_{0} = x_{n}.

14.2.1.1 Given an oriented edge e of X, let f_{e} be the positive linear map of [0,1] onto e; it is a path from the initial point of e to the final point of e. Then, corresponding to the edge path e_{1}, .., e_{n} from x_{0} to x_{n}, one has the actual path f = f_{1} * (f_{2} * (\cdots * f_{n})) from x_{0} to x_{n}, where f_{i} = f_{e_{i}}; it is uniquely determined by the edge path e_{1}, .., e_{n}. We call it the path corresponding to the edge path e_{1}, .., e_{n}. If the edge path is closed, then the corresponding path f is a loop.

14.2.2 (Lem 84.1) A graph X is connected iff every pair of vertices of X can be joined by an edge path in X.

14.2.3 Let e_{1}, …, e_{n} be an edge path in the linear graph X. It can happen that for some i, the oriented edges e_{i} and e_{i+1} consist of the same edge of X, but with opposite orientations. If this situation does not occur, then the edge path is said to be a reduced edge path.

14.2.4 A subgraph T of X is said to be a tree in X if T is connected and T contains no closed reduced edge paths.

14.2.5 (Lem 84.2) If T is a tree in X, and if A is an edge of X that intersects T in a single vertex, then T \cup A is a tree in X. Conversely, if T is a finite tree in X that consists of more than one edge, then there is a tree T_{0} in X and an edge A of X that intersects T_{0} in a single vertex, such that T = T_{0} \cup A.

14.2.6 (Thm 84.3) Any tree T is simply connected.

14.2.7 A tree T in X is maximal if there is no tree in X that properly contains T.

14.2.8 (Thm 84.4) Let X be a connected graph. A tree T in X is maximal iff it contains all the vertices of X.

14.2.9 (Thm 84.5) If X is a linear graph, every tree T_{0} in X is contained in a maximal tree in X.

14.2.10 (Lem 84.6) Suppose X = U \cup V, where U and V are open sets of X. Suppose that U \cap V is the union of two disjoint open path connected sets A and B, that \alpha is path in U from the point a of A to the point b of B, and that \beta is a path in V from b to a. If U and V are simply connected, then the class [\alpha * \beta ] generates \pi_{1}(X,a).

14.2.11 (Thm 84.7) Let X be a connected graph that is not a tree, Then the fundamental group of X is an nontrivial free group, Indeed, if T is a maximal tree in X, then the fundamental group of X has a system of free generators that is in bijective correspondence with the collection of edges of X that are not in T.

14.3 Subgroups of Free groups

14.3.1 (Thm 85.1) If H is a subgroup of a free group F, then H is free.

14.3.2 If X is a finite linear graph, we define the Euler number of X to be the number of vertices of X minus the number of edges. It is commonly denoted by \chi (X).

14.3.3 (Lem 85.2) If X is a finite, connected linear graph, then the cardinality of a system of free generators for the fundamental group of X is 1- \chi (X).

14.3.4 Let H be a subgroup of the group G. If the collection G?H of right cosets of H in G is finite, its cardinality is called the index of H in G.

14.3.4.1 The collection of left cosets of H in G h as the same cardinality as it.

14.3.5 (Thm 85.3) Let F be a gree group with n+1 free generators; let H be a subgroup of F. If H has index k in F, then H has kn+1 free generators.

Contact

Seoul Nat’l Univ Computer Science & Engineering Graduate

Human-centered Computer Systems Lab

Ph.D. Student of Youngki Lee

Email : ohsai@snu.ac.kr

Github : ohsai

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