Measure-Theoretic Calculus in Abstract Spaces: On the Playground of Infinite-Dimensional Spaces

✍ Scribed by Zigang Pan

• Publisher: Birkhäuser
• Year: 2024
• Tongue: English
• Leaves: 952
• Edition: 1st ed. 2023
• Category: Library

✦ Synopsis

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.

✦ Table of Contents

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