Topics In Analysis

52.1.2 Rouche’s Theorem

With the argument principle, it is possible to prove Rouche’s theorem . In the argument principle, denote by Zf the quantity ∑ k=1mrk and by Pf the quantity ∑ k=1nlk. Thus Zf is the number of zeros of f counted according to the order of the zero with a similar definition holding for Pf. Thus the conclusion of the argument principle is.

Rouche’s theorem allows the comparison of Zh − Ph for h = f,g. It is a wonderful and amazing result.

Theorem 52.1.4 (Rouche’s theorem)Let f,g be meromorphic in an open set Ω. Also suppose γ∗ is a closed bounded variation curve with the property that for all z

Ω,n
= 0, no zeros or poles are on γ∗, and for all z ∈ Ω,n
either equals 0 or 1. Let Zf and Pf denote respectively the numbers of zeros and poles of f, which have the property that the winding number equals 1, counted according to order, with Zg and Pg being defined similarly. Also suppose that for z ∈ γ∗

(52.1.4)

Then

Proof: From the hypotheses,

which shows that for all z ∈ γ∗,

Letting l denote a branch of the logarithm defined on ℂ ∖ [0,∞), it follows that l

is a primitive for the function,

Therefore, by the argument principle,

This proves the theorem.

Often another condition other than 52.1.4 is used.

Corollary 52.1.5 In the situation of Theorem 52.1.4 change 52.1.4 to the condition,

for z ∈ γ∗. Then the conclusion is the same.

Proof: The new condition implies

<
on γ∗. Therefore,
(−∞,0] and so you can do the same argument with a branch of the logarithm.

Part August 30, 2018 I  Review Of Advanced Calculus

Chapter 1 Set Theory

1.1 Basic Definitions
1.2 The Schroder Bernstein Theorem
1.3 Equivalence Relations
1.4 Partially Ordered Sets

Chapter 2 Continuous Functions Of One Variable

2.1 Exercises
2.2 Theorems About Continuous Functions

Chapter 3 The Riemann Stieltjes Integral

3.1 Upper And Lower Riemann Stieltjes Sums
3.2 Exercises
3.3 Functions Of Riemann Integrable Functions
3.4 Properties Of The Integral
3.5 Fundamental Theorem Of Calculus
3.6 Exercises

Chapter 4 Some Important Linear Algebra

4.1 Algebra In Fn
4.2 Exercises
4.3 The Inner Product And Distance In ℂn
4.4 Subspaces Spans And Bases
4.5 An Application To Matrices
4.6 The Mathematical Theory Of Determinants
4.6.1 The Function sgn
4.7 The Determinant
4.7.1 The Definition
4.7.2 Permuting Rows Or Columns
4.7.3 A Symmetric Definition
4.7.4 The Alternating Property Of The Determinant
4.7.5 Linear Combinations And Determinants
4.7.6 The Determinant Of A Product
4.7.7 Cofactor Expansions
4.7.8 Formula For The Inverse
4.7.9 Cramer’s Rule
4.7.10 Upper Triangular Matrices
4.8 The Cayley Hamilton Theorem∗
4.9 An Identity Of Cauchy
4.10 Block Multiplication Of Matrices
4.11 Exercises
4.12 Shur’s Theorem
4.13 The Right Polar Decomposition

Chapter 5 Multi-variable Calculus

5.1 Continuous Functions
5.2 Open And Closed Sets
5.3 Continuous Functions
5.3.1 Sufficient Conditions For Continuity
5.4 Exercises
5.5 Limits Of A Function
5.6 Exercises
5.7 The Limit Of A Sequence
5.7.1 Sequences And Completeness
5.7.2 Continuity And The Limit Of A Sequence
5.8 Properties Of Continuous Functions
5.9 Exercises
5.10 Proofs Of Theorems
5.11 The Space ℒ(Fn, Fm)
5.11.1 The Operator Norm
5.12 The Frechet Derivative
5.13 C1 Functions
5.14 Ck Functions
5.15 Mixed Partial Derivatives
5.16 Implicit Function Theorem
5.16.1 More Continuous Partial Derivatives
5.17 The Method Of Lagrange Multipliers

Chapter 6 Metric Spaces And General Topological Spaces

6.1 Metric Space
6.2 Compactness In Metric Space
6.3 Some Applications Of Compactness
6.4 Ascoli Arzela Theorem
6.5 Another General Version
6.6 The Tietze Extension Theorem
6.7 Some Simple Fixed Point Theorems
6.8 General Topological Spaces
6.9 Connected Sets
6.10 Exercises

Chapter 7 Weierstrass Approximation Theorem

7.1 The Bernstein Polynomials
7.2 Stone Weierstrass Theorem
7.2.1 The Case Of Compact Sets
7.2.2 The Case Of Locally Compact Sets
7.2.3 The Case Of Complex Valued Functions
7.3 The Holder Spaces
7.4 Exercises

Chapter 8 Brouwer Fixed Point Theorem ℝn∗

8.1 Simplices And Triangulations
8.2 Labeling Vertices
8.3 The Brouwer Fixed Point Theorem
8.4 Invariance Of Domain

Part II  Real And Abstract Analysis

Chapter 9 Abstract Measure And Integration

9.1 σ Algebras
9.2 Exercises
9.3 The Abstract Lebesgue Integral
9.3.1 Preliminary Observations
9.3.2 The Lebesgue Integral Nonnegative Functions
9.3.3 The Lebesgue Integral For Nonnegative Simple Functions
9.3.4 Simple Functions And Measurable Functions
9.3.5 The Monotone Convergence Theorem
9.3.6 Other Definitions
9.3.7 Fatou’s Lemma
9.3.8 The Righteous Algebraic Desires Of The Lebesgue Integral
9.4 The Space L1
9.5 Vitali Convergence Theorem
9.6 Exercises

Chapter 10 The Construction Of Measures

10.1 Outer Measures
10.2 Urysohn’s lemma
10.3 Positive Linear Functionals
10.4 One Dimensional Lebesgue Measure
10.5 One Dimensional Lebesgue Stieltjes Measure
10.6 The Distribution Function
10.7 Good Lambda Inequality
10.8 The Ergodic Theorem
10.9 Product Measures
10.10 Alternative Treatment Of Product Measure
10.10.1 Monotone Classes And Algebras
10.10.2 Product Measure
10.11 Completion Of Measures
10.12 Another Version Of Product Measures
10.12.1 General Theory
10.12.2 Completion Of Product Measure Spaces
10.13 Disturbing Examples
10.14 Exercises

Chapter 11 Lebesgue Measure

11.1 Basic Properties
11.2 The Vitali Covering Theorem
11.3 The Vitali Covering Theorem (Elementary Version)
11.4 Vitali Coverings
11.5 Change Of Variables For Linear Maps
11.6 Change Of Variables For C1 Functions
11.7 Mappings Which Are Not One To One
11.8 Lebesgue Measure And Iterated Integrals
11.9 Spherical Coordinates In p Dimensions
11.10 The Brouwer Fixed Point Theorem
11.11 The Brouwer Fixed Point Theorem Another Proof
11.12 Invariance Of Domain

Chapter 12 Some Extension Theorems

12.1 Algebras
12.2 Caratheodory Extension Theorem
12.3 The Tychonoff Theorem
12.4 Kolmogorov Extension Theorem
12.5 Exercises

Chapter 13 The Lp Spaces

13.1 Basic Inequalities And Properties
13.2 Density Considerations
13.3 Separability
13.4 Continuity Of Translation
13.5 Mollifiers And Density Of Smooth Functions
13.6 Exercises

Chapter 14 Stone’s Theorem And Partitions Of Unity

14.1 Partitions Of Unity And Stone’s Theorem
14.2 An Extension Theorem, Retracts
14.3 Something Which Is Not A Retract
14.4 Exercises

Chapter 15 Banach Spaces

15.1 Theorems Based On Baire Category
15.1.1 Baire Category Theorem
15.1.2 Uniform Boundedness Theorem
15.1.3 Open Mapping Theorem
15.1.4 Closed Graph Theorem
15.2 Hahn Banach Theorem
15.2.1 Partially Ordered Sets
15.2.2 Gauge Functions And Hahn Banach Theorem
15.2.3 The Complex Version Of The Hahn Banach Theorem
15.2.4 The Dual Space And Adjoint Operators
15.3 Uniform Convexity Of Lp
15.4 Closed Subspaces
15.5 Weak And Weak ∗ Topologies
15.5.1 Basic Definitions
15.5.2 Banach Alaoglu Theorem
15.5.3 Eberlein Smulian Theorem
15.6 Operators With Closed Range
15.7 Exercises

Chapter 16 Locally Convex Topological Vector Spaces

16.1 Fundamental Considerations
16.2 Separation Theorems
16.2.1 Convex Functionals
16.2.2 More Separation Theorems
16.3 The Weak And Weak∗ Topologies
16.4 Mean Ergodic Theorem
16.5 The Tychonoff And Schauder Fixed Point Theorems
16.6 A Variational Principle of Ekeland
16.6.1 Cariste Fixed Point Theorem
16.6.2 A Density Result
16.7 Quotient Spaces

Chapter 17 Hilbert Spaces

17.1 Basic Theory
17.2 The Hilbert Space L(U)
17.3 Approximations In Hilbert Space
17.4 The Müntz Theorem
17.5 Orthonormal Sets
17.6 Fourier Series, An Example
17.7 Compact Operators
17.7.1 Compact Operators In Hilbert Space
17.8 Sturm Liouville Problems
17.8.1 Nuclear Operators
17.8.2 Hilbert Schmidt Operators
17.9 Compact Operators In Banach Space
17.10 The Fredholm Alternative
17.11 Square Roots
17.12 Ordinary Differential Equations in Banach Space
17.13 Fractional Powers of Operators
17.14 General Theory Of Continuous Semigroups
17.14.1 An Evolution Equation
17.14.2 Adjoints, Hilbert Space
17.14.3 Adjoints, Reflexive Banach Space

Chapter 18 Representation Theorems

18.1 Radon Nikodym Theorem
18.2 Vector Measures
18.3 Representation Theorems For The Dual Space Of Lp
18.4 The Dual Space Of L∞(Ω)
18.5 Non σ Finite Case
18.6 The Dual Space Of C0 (X)
18.7 The Dual Space Of C0(X), Another Approach
18.8 More Attractive Formulations
18.9 Sequential Compactness In L1
18.10 Exercises

Chapter 19 The Bochner Integral

19.1 Strong And Weak Measurability
19.2 The Bochner Integral
19.2.1 Definition And Basic Properties
19.2.2 Taking A Closed Operator Out Of The Integral
19.3 Operator Valued Functions
19.3.1 Review Of Hilbert Schmidt Theorem
19.3.2 Measurable Compact Operators
19.4 Fubini’s Theorem For Bochner Integrals
19.5 The Spaces Lp (Ω;X)
19.6 Measurable Representatives
19.7 Vector Measures
19.8 The Riesz Representation Theorem
19.9 Pointwise Behavior Of Weakly Convergent Sequences
19.10 Exercises

Chapter 20 The Derivative

20.1 Limits Of A Function
20.2 Basic Definitions
20.3 The Chain Rule
20.4 The Derivative Of A Compact Mapping
20.5 The Matrix Of The Derivative
20.6 A Mean Value Inequality
20.7 Higher Order Derivatives
20.8 The Derivative And The Cartesian Product
20.9 Mixed Partial Derivatives
20.10 Implicit Function Theorem
20.11 More Derivatives
20.12 Lyapunov Schmidt Procedure
20.13 Analytic Functions
20.14 Ordinary Differential Equations
20.15 Exercises

Chapter 21 Degree Theory, An Introduction

21.1 Sard’s Lemma and Approximation
21.2 An Identity and Surprising Implications
21.3 Definitions And Elementary Properties
21.4 Borsuk’s Theorem
21.5 Applications
21.6 The Product Formula and Jordan Separation Theorem
21.7 Jordan Curve Theorem
21.8 A Function With Values In Smaller Dimensions
21.9 The Leray Schauder Degree
21.10 Exercises

Chapter 22 Critical Points

22.1 Mountain Pass Theorem In Hilbert Space
22.1.1 A Locally Lipschitz Selection, Pseudogradients
22.1.2 Mountain Pass Theorem In Banach Space

Chapter 23 Nonlinear Operators

23.1 Some Nonlinear Single Valued Operators
23.2 Duality Maps
23.3 Penalizaton And Projection Operators
23.4 Set-Valued Maps, Pseudomonotone Operators
23.5 Sum Of Pseudomonotone Operators
23.6 Generalized Gradients
23.7 Maximal Monotone Operators
23.7.1 The minmax Theorem
23.7.2 Equivalent Conditions For Maximal Monotone
23.7.3 Surjectivity Theorems
23.7.4 Approximation Theorems
23.7.5 Sum Of Maximal Monotone Operators
23.7.6 Convex Functions, An Example
23.8 Perturbation Theorems

Chapter 24 Integrals And Derivatives

24.1 The Fundamental Theorem Of Calculus
24.2 Absolutely Continuous Functions
24.3 Weak Derivatives
24.4 Lipschitz Functions
24.5 Rademacher’s Theorem First Version
24.6 Rademacher’s Theorem
24.6.1 Morrey’s Inequality
24.6.2 Rademacher’s Theorem
24.7 Differentiation Of Measures With Respect To Lebesgue Measure
24.8 Exercises

Chapter 25 Orlitz Spaces

25.1 Basic Theory
25.2 Dual Spaces In Orlitz Space

Chapter 26 Hausdorff Measure

26.1 Definition Of Hausdorff Measures
26.1.1 Properties Of Hausdorff Measure
26.2 ℋn And mn
26.3 Technical Considerations
26.3.1 Steiner Symmetrization
26.3.2 The Isodiametric Inequality
26.4 The Proper Value Of β (n)
26.4.1 A Formula For α (n)
26.4.2 Hausdorff Measure And Linear Transformations

Chapter 27 The Area Formula

27.0.1 Preliminary Results
27.0.2 The Area Formula
27.1 Mappings That Are Not One To One
27.2 The Divergence Theorem
27.3 Integration And The Degree
27.4 The Case Of W1,p

Chapter 28 Integration Of Differential Forms

28.1 Manifolds
28.2 The Binet Cauchy Formula
28.3 Integration Of Differential Forms On Manifolds
28.3.1 The Derivative Of A Differential Form
28.4 Stoke’s Theorem And The Orientation Of ∂Ω
28.5 Green’s Theorem
28.5.1 An Oriented Manifold
28.5.2 Green’s Theorem
28.6 The Divergence Theorem

Chapter 29 Differentiation Of Radon Measures

29.1 Besicovitch Covering Theorem
29.2 Fundamental Theorem Of Calculus For Radon Measures
29.3 Slicing Measures
29.4 Vitali Coverings
29.5 Differentiation Of Radon Measures
29.6 The Radon Nikodym Theorem For Radon Measures

Chapter 30 Fourier Transforms

30.1 An Algebra Of Special Functions
30.2 Fourier Transforms Of Functions In G
30.3 Fourier Transforms Of Just About Anything
30.3.1 Fourier Transforms Of G∗
30.3.2 Fourier Transforms Of Functions In L1 (ℝn)
30.3.3 Fourier Transforms Of Functions In L2 (ℝn)
30.3.4 The Schwartz Class
30.3.5 Convolution
30.4 Exercises

Chapter 31 Fourier Analysis In ℝn An Introduction

31.1 The Marcinkiewicz Interpolation Theorem
31.2 The Calderon Zygmund Decomposition
31.3 Mihlin’s Theorem
31.4 Singular Integrals
31.5 Helmholtz Decompositions

Chapter 32 Gelfand Triples And Related Stuff

32.1 An Unnatural Example
32.2 Standard Techniques In Evolution Equations
32.3 An Important Formula
32.4 The Implicit Case
32.5 The Implicit Case, B = B (t)
32.6 Another Approach
32.7 Some Imbedding Theorems
32.8 Some Evolution Inclusions

Chapter 33 Maximal Monotone Operators, Hilbert Space

33.1 Basic Theory
33.2 Evolution Inclusions
33.3 Subgradients
33.3.1 General Results
33.3.2 Hilbert Space
33.4 A Perturbation Theorem
33.5 An Evolution Inclusion
33.6 A More Complicated Perturbation Theorem
33.7 An Evolution Inclusion

Part III  Sobolev Spaces

Chapter 34 Weak Derivatives

34.1 Weak ∗ Convergence
34.2 Test Functions And Weak Derivatives
34.3 Weak Derivatives In Llocp
34.4 Morrey’s Inequality
34.5 Rademacher’s Theorem
34.6 Change Of Variables Formula Lipschitz Maps

Chapter 35 The Area And Coarea Formulas

35.1 The Area Formula Again
35.2 Mappings That Are Not One To One
35.3 The Coarea Formula
35.4 A Nonlinear Fubini’s Theorem

Chapter 36 Integration On Manifolds

36.1 Partitions Of Unity
36.2 Integration On Manifolds
36.3 Comparison With ℋn

Chapter 37 Basic Theory Of Sobolev Spaces

37.1 Embedding Theorems For Wm,p (ℝn)
37.2 An Extension Theorem
37.3 General Embedding Theorems
37.4 More Extension Theorems

Chapter 38 Sobolev Spaces Based On L2

38.1 Fourier Transform Techniques
38.2 Fractional Order Spaces
38.3 An Intrinsic Norm
38.4 Embedding Theorems
38.5 The Trace On The Boundary Of A Half Space
38.6 Sobolev Spaces On Manifolds
38.6.1 General Theory
38.6.2 The Trace On The Boundary

Chapter 39 Weak Solutions

39.1 The Lax Milgram Theorem
39.2 An Application Of The Mountain Pass Theorem

Chapter 40 Korn’s Inequality

40.1 A Fundamental Inequality
40.2 Korn’s Inequality

Chapter 41 Elliptic Regularity And Nirenberg Differences

41.1 The Case Of A Half Space
41.2 The Case Of Bounded Open Sets

Chapter 42 Interpolation In Banach Space

42.1 Some Standard Techniques In Evolution Equations
42.1.1 Weak Vector Valued Derivatives
42.2 An Important Formula
42.3 The Implicit Case
42.4 Some Implicit Inclusions
42.5 Some Imbedding Theorems
42.6 The K Method
42.7 The J Method
42.8 Duality And Interpolation

Chapter 43 Trace Spaces

43.1 Definition And Basic Theory Of Trace Spaces
43.2 Trace And Interpolation Spaces

Chapter 44 Traces Of Sobolev Spaces And Fractional Order Spaces

44.1 Traces Of Sobolev Spaces On The Boundary Of A Half Space
44.2 A Right Inverse For The Trace For A Half Space
44.3 Intrinsic Norms
44.4 Fractional Order Sobolev Spaces

Chapter 45 Sobolev Spaces On Manifolds

45.1 Basic Definitions
45.2 The Trace On The Boundary Of An Open Set

Part IV  Multifunctions

Chapter 46 The Yankov von Neumann Aumann theorem

Chapter 47 Multifunctions And Their Measurability

47.1 The General Case
47.1.1 A Special Case Which Is Easier
47.1.2 Other Measurability Considerations
47.2 Existence Of Measurable Fixed Points
47.2.1 Simplices And Labeling
47.2.2 Labeling Vertices
47.2.3 Measurability Of Brouwer Fixed Points
47.2.4 Measurability Of Schauder Fixed Points
47.3 A Set Valued Browder Lemma With Measurability
47.4 A Measurable Kakutani Theorem
47.5 Some Variational Inequalities
47.6 An Example
47.7 Limit Conditions For Nemytskii Operators

Part V  Complex Analysis

Chapter 48 The Complex Numbers

48.1 The Extended Complex Plane
48.2 Exercises

Chapter 49 Riemann Stieltjes Integrals

49.1 Exercises

Chapter 50 Fundamentals Of Complex Analysis

50.1 Analytic Functions
50.1.1 Cauchy Riemann Equations
50.1.2 An Important Example
50.2 Exercises
50.3 Cauchy’s Formula For A Disk
50.4 Exercises
50.5 Zeros Of An Analytic Function
50.6 Liouville’s Theorem
50.7 The General Cauchy Integral Formula
50.7.1 The Cauchy Goursat Theorem
50.7.2 A Redundant Assumption
50.7.3 Classification Of Isolated Singularities
50.7.4 The Cauchy Integral Formula
50.7.5 An Example Of A Cycle
50.8 Exercises

Chapter 51 The Open Mapping Theorem

51.1 A Local Representation
51.2 Branches Of The Logarithm
51.3 Maximum Modulus Theorem
51.4 Extensions Of Maximum Modulus Theorem
51.4.1 Phragmên Lindelöf Theorem
51.4.2 Hadamard Three Circles Theorem
51.4.3 Schwarz’s Lemma
51.4.4 One To One Analytic Maps On The Unit Ball
51.5 Exercises
51.6 Counting Zeros
51.7 An Application To Linear Algebra
51.8 Exercises

Chapter 52 Residues

52.1 Rouche’s Theorem And The Argument Principle
52.1.1 Argument Principle
52.1.2 Rouche’s Theorem
52.1.3 A Different Formulation
52.2 Singularities And The Laurent Series
52.2.1 What Is An Annulus?
52.2.2 The Laurent Series
52.2.3 Contour Integrals And Evaluation Of Integrals
52.3 Exercises

Chapter 53 Some Important Functional Analysis Applications

53.1 The Spectral Radius Of A Bounded Linear Transformation
53.2 Analytic Semigroups
53.2.1 Sectorial Operators And Analytic Semigroups
53.2.2 The Numerical Range
53.2.3 An Interesting Example
53.2.4 Fractional Powers Of Sectorial Operators
53.2.5 A Scale Of Banach Spaces

Chapter 54 Complex Mappings

54.1 Conformal Maps
54.2 Fractional Linear Transformations
54.2.1 Circles And Lines
54.2.2 Three Points To Three Points
54.3 Riemann Mapping Theorem
54.3.1 Montel’s Theorem
54.3.2 Regions With Square Root Property
54.4 Analytic Continuation
54.4.1 Regular And Singular Points
54.4.2 Continuation Along A Curve
54.5 The Picard Theorems
54.5.1 Two Competing Lemmas
54.5.2 The Little Picard Theorem
54.5.3 Schottky’s Theorem
54.5.4 A Brief Review
54.5.5 Montel’s Theorem
54.5.6 The Great Big Picard Theorem
54.6 Exercises

Chapter 55 Approximation By Rational Functions

55.1 Runge’s Theorem
55.1.1 Approximation With Rational Functions
55.1.2 Moving The Poles And Keeping The Approximation
55.1.3 Merten’s Theorem.
55.1.4 Runge’s Theorem
55.2 The Mittag-Leffler Theorem
55.2.1 A Proof From Runge’s Theorem
55.2.2 A Direct Proof Without Runge’s Theorem
55.2.3 Functions Meromorphic On
55.2.4 Great And Glorious Theorem, Simply Connected Regions
55.3 Exercises

Chapter 56 Infinite Products

56.1 Analytic Function With Prescribed Zeros
56.2 Factoring A Given Analytic Function
56.2.1 Factoring Some Special Analytic Functions
56.3 The Existence Of An Analytic Function With Given Values
56.4 Jensen’s Formula
56.5 Blaschke Products
56.5.1 The Müntz-Szasz Theorem Again
56.6 Exercises

Chapter 57 Elliptic Functions

57.1 Periodic Functions
57.1.1 The Unimodular Transformations
57.1.2 The Search For An Elliptic Function
57.1.3 The Differential Equation Satisfied By ℘
57.1.4 A Modular Function
57.1.5 A Formula For λ
57.1.6 Mapping Properties Of λ
57.1.7 A Short Review And Summary
57.2 The Picard Theorem Again
57.3 Exercises

Part VI  Topics In Probability

Chapter 58 Basic Probability

58.1 Random Variables And Independence
58.2 Kolmogorov Extension Theorem For Polish Spaces
58.3 Independence
58.4 Independence For Banach Space Valued Random Variables
58.5 Reduction To Finite Dimensions
58.6 0,1 Laws
58.7 Kolmogorov’s Inequality, Strong Law Of Large Numbers
58.8 The Characteristic Function
58.9 Conditional Probability
58.10 Conditional Expectation
58.11 Characteristic Functions And Independence
58.12 Characteristic Functions For Measures
58.13 Characteristic Functions And Independence In Banach Space
58.14 Convolution And Sums
58.15 The Convergence Of Sums Of Symmetric Random Variables
58.16 The Multivariate Normal Distribution
58.17 Use Of Characteristic Functions To Find Moments
58.18 The Central Limit Theorem
58.19 Characteristic Functions, Prokhorov Theorem
58.20 Generalized Multivariate Normal
58.21 Positive Definite Functions, Bochner’s Theorem

Chapter 59 Conditional Expectation And Martingales

59.1 Conditional Expectation
59.2 Discrete Martingales
59.2.1 Upcrossings
59.2.2 The Submartingale Convergence Theorem
59.2.3 Doob Submartingale Estimate
59.3 Optional Sampling And Stopping Times
59.3.1 Stopping Times And Their Properties
59.4 Optional Stopping Times And Martingales
59.4.1 Stopping Times And Their Properties
59.5 Submartingale Convergence Theorem
59.5.1 Upcrossings
59.5.2 Maximal Inequalities
59.5.3 The Upcrossing Estimate
59.6 The Submartingale Convergence Theorem
59.7 A Reverse Submartingale Convergence Theorem
59.8 Strong Law Of Large Numbers

Chapter 60 Probability In Infinite Dimensions

60.1 Conditional Expectation In Banach Spaces
60.2 Probability Measures And Tightness
60.3 Tight Measures
60.4 A Major Existence And Convergence Theorem
60.5 Bochner’s Theorem In Infinite Dimensions
60.6 The Multivariate Normal Distribution
60.7 Gaussian Measures
60.7.1 Definitions And Basic Properties
60.7.2 Fernique’s Theorem
60.8 Gaussian Measures For A Separable Hilbert Space
60.9 Abstract Wiener Spaces
60.10 White Noise
60.11 Existence Of Abstract Wiener Spaces

Chapter 61 Stochastic Processes

61.1 Fundamental Definitions And Properties
61.2 Kolmogorov Čentsov Continuity Theorem
61.3 Filtrations
61.4 Martingales
61.5 Some Maximal Estimates
61.6 Optional Sampling Theorems
61.6.1 Stopping Times And Their Properties
61.6.2 Doob Optional Sampling Theorem
61.7 Doob Optional Sampling Continuous Case
61.7.1 Stopping Times
61.7.2 The Optional Sampling Theorem Continuous Case
61.8 Right Continuity Of Submartingales
61.9 Some Maximal Inequalities
61.10 Continuous Submartingale Convergence Theorem
61.11 Hitting This Before That
61.12 The Space ℳTp (E)

Chapter 62 The Quadratic Variation Of A Martingale

62.1 How To Recognize A Martingale
62.2 The Quadratic Variation
62.3 The Covariation
62.4 The Burkholder Davis Gundy Inequality
62.5 The Quadratic Variation And Stochastic Integration
62.6 Another Limit For Quadratic Variation
62.7 Doob Meyer Decomposition
62.8 Levy’s Theorem

Chapter 63 Wiener Processes

63.1 Real Wiener Processes
63.2 Nowhere Differentiability Of Wiener Processes
63.3 Wiener Processes In Separable Banach Space
63.4 An Example Of Martingales, Independent Increments
63.5 Hilbert Space Valued Wiener Processes
63.6 Wiener Processes, Another Approach
63.6.1 Lots Of Independent Normally Distributed Random Variables
63.6.2 The Wiener Processes
63.6.3 Q Wiener Processes In Hilbert Space
63.6.4 Levy’s Theorem In Hilbert Space

Chapter 64 Stochastic Integration

64.1 Integrals Of Elementary Processes
64.2 Different Definition Of Elementary Functions
64.3 Approximating With Elementary Functions
64.4 Some Hilbert Space Theory
64.5 The General Integral
64.6 The Case That Q Is Trace Class
64.7 A Short Comment On Measurability
64.8 Localization For Elementary Functions
64.9 Localization In General
64.10 The Stochastic Integral As A Local Martingale
64.11 The Quadratic Variation Of The Stochastic Integral
64.12 The Holder Continuity Of The Integral
64.13 Taking Out A Linear Transformation
64.14 A Technical Integration By Parts Result
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