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

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. 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