Measure-Theoretic Calculus in Abstract Spaces : On the Playground of Infinite-Dimensional Spaces / Zigang Pan.

Format:

Book

Author/Creator:

Pan, Zigang, author.

Language:

English

Subjects (All):

Measure theory.

Automatic control.

Functional analysis.

Physical Description:

1 online resource (951 pages)

Edition:

First edition.

Place of Publication:

Cham, Switzerland : Birkhäuser, [2023]

Summary:

This monograph provides a rigorous, encyclopedic treatment of the fundamental topics in real analysis, functional analysis, and measure theory. The result of many years of the author’s careful and extensive work, this text synthesizes and builds upon the existing literature in an effort to develop and solidify the theory of measure-theoretic calculus in abstract spaces. Standard results and proofs are illustrated in general abstract settings under rigorous treatment, and numerous ancillary topics are also covered in detail, such as functional analytic treatment of optimization, probability theory, and the theory of Sobolev spaces. Applied mathematicians and researchers working in control theory, operations research, economics, optimization theory, and many other areas will find this text to be a comprehensive and invaluable resource. It can also serve as an analysis textbook for graduate-level students.

Contents:

Intro

Preface

Contents

List of Figures

Notations

1 Introduction

1.1 The Tour of the Book

1.2 How to Use the Book

1.3 What This Book Does Not Include

2 Set Theory

2.1 Axiomatic Foundations of Set Theory

2.2 Relations and Equivalence

2.3 Function

2.4 Set Operations

2.5 Algebra of Sets

2.6 Partial Ordering and Total Ordering

2.7 Basic Principles

3 Topological Spaces

3.1 Fundamental Notions

3.2 Continuity

3.3 Basis and Countability

3.4 Products of Topological Spaces

3.5 The Separation Axioms

3.6 Category Theory

3.7 Connectedness

3.8 Continuous Real-Valued Functions

3.9 Nets and Convergence

4 Metric Spaces

4.1 Fundamental Notions

4.2 Convergence and Completeness

4.3 Uniform Continuity and Uniformity

4.4 Product Metric Spaces

4.5 Subspaces

4.6 Baire Category

4.7 Completion of Metric Spaces

4.8 Metrization of Topological Spaces

4.9 Interchange Limits

5 Compact and Locally Compact Spaces

5.1 Compact Spaces

5.2 Countable and Sequential Compactness

5.3 Real-Valued Functions and Compactness

5.4 Compactness in Metric Spaces

5.5 The Ascoli-Arzelá Theorem

5.6 Product Spaces

5.7 Locally Compact Spaces

5.7.1 Fundamental Notion

5.7.2 Partition of Unity

5.7.3 The Alexandroff One-point Compactification

5.7.4 Proper Functions

5.8 σ-Compact Spaces

5.9 Paracompact Spaces

5.10 The Stone-Čech Compactification

6 Vector Spaces

6.1 Group

6.2 Ring

6.3 Field

6.4 Vector Spaces

6.5 Product Spaces

6.6 Subspaces

6.7 Convex Sets

6.8 Linear Independence and Dimensions

7 Banach Spaces

7.1 Normed Linear Spaces

7.2 The Natural Metric

7.3 Product Spaces

7.4 Banach Spaces

7.5 Compactness

7.6 Quotient Spaces

7.7 The Stone-Weierstrass Theorem

7.8 Linear Operators

7.9 Dual Spaces.

7.9.1 Basic Concepts

7.9.2 Duals of Some Common Banach Spaces

7.9.3 Extension Form of Hahn-Banach Theorem

7.9.4 Second Dual Space

7.9.5 Alignment and Orthogonal Complements

7.10 The Open Mapping Theorem

7.11 The Adjoints of Linear Operators

7.12 Weak Topology

8 Global Theory of Optimization

8.1 Hyperplanes and Convex Sets

8.2 Geometric Form of Hahn-Banach Theorem

8.3 Duality in Minimum Norm Problems

8.4 Convex and Concave Functionals

8.5 Conjugate Convex Functionals

8.6 Fenchel Duality Theorem

8.7 Positive Cones and Convex Mappings

8.8 Lagrange Multipliers

9 Differentiation in Banach Spaces

9.1 Fundamental Notion

9.2 The Derivatives of Some Common Functions

9.3 Chain Rule and Mean Value Theorem

9.4 Higher Order Derivatives

9.4.1 Basic Concept

9.4.2 Interchange Order of Differentiation

9.4.3 High Order Derivatives of Some Common Functions

9.4.4 Properties of High Order Derivatives

9.5 Mapping Theorems

9.6 Global Inverse Function Theorem

9.7 Interchange Differentiation and Limit

9.8 Tensor Algebra

9.9 Analytic Functions

9.10 Newton's Method

10 Local Theory of Optimization

10.1 Basic Notion

10.2 Unconstrained Optimization

10.3 Optimization with Equality Constraints

10.4 Inequality Constraints

11 General Measure and Integration

11.1 Measure Spaces

11.2 Outer Measure and the Extension Theorem

11.3 Measurable Functions

11.4 Integration

11.5 General Convergence Theorems

11.6 Banach Space Valued Measures

11.7 Calculation with Measures

11.8 The Radon-Nikodym Theorem

11.9 Lp Spaces

11.10 Dual of C(X,Y) and Cc(X,Y)

12 Differentiation and Integration

12.1 Carathéodory Extension Theorem

12.2 Change of Variable

12.3 Product Measure

12.4 Functions of Bounded Variation

12.5 Absolute and Lipschitz Continuity.

12.6 Fundamental Theorem of Calculus

12.7 Representation of (Ck(Ω,Y))*

12.8 Sobolev Spaces

12.9 Integral Depending on a Parameter

12.10 Iterated Integrals

12.11 Manifold

12.11.1 Basic Notion

12.11.2 Tangent Vectors

12.11.3 Vector Fields

13 Hilbert Spaces

13.1 Fundamental Notions

13.2 Projection Theorems

13.3 Dual of Hilbert Spaces

13.4 Hermitian Adjoints

13.5 Approximation in Hilbert Spaces

13.6 Other Minimum Norm Problems

13.7 Positive Definite Operators on Hilbert Spaces

13.8 Pseudoinverse Operator

13.9 Spectral Theory of Linear Operators

14 Probability Theory

14.1 Fundamental Notions

14.2 Gaussian Random Variables and Vectors

14.3 Law of Large Numbers

14.4 Martingales Indexed by Z+

14.5 Banach Space Valued Martingales Indexed by Z+

14.6 Characteristic Functions

14.7 Convergence in Distribution

14.8 Central Limit Theorem

14.9 Uniform Integrability and Martingales

14.10 Existence of the Wiener Process

14.11 Martingales with General Index Set

14.12 Stochastic Integral

14.13 Itô Processes

14.14 Girsanov's Theorem

A Elements in Calculus

A.1 Some Formulas

A.2 Convergence of Infinite Sequences

A.3 Riemann-Stieltjes Integral

Bibliography

Index.

Notes:

Includes bibliographical references and index.

Description based on print version record.

Other Format:

Print version: Pan, Zigang Measure-Theoretic Calculus in Abstract Spaces

ISBN:

3-031-21912-0