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

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