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1 Metric Spaces 1.1 Introduction 1.1.1 Basic Definitions and Notation 1.1.2 Sequences and Complete Metric Spaces 1.1.3 Topology of Metric Spaces 1.1.4 Baire Theorem 1.1.5 Continuous and Uniformly Continuous Functions 1.1.6 Completion of Metric Spaces: Equivalence of Metrics 1.1.7 Pointwise and Uniform Convergence of Maps 1.1.8 Compact Metric Spaces 1.1.9 Connectedness 1.1.10 Partitions of Unity 1.1.11 Products of Metric Spaces 1.1.12 Auxiliary Notions 1.2 Problems 1.3 Solutions Bibliography 2 Topological Spaces 2.1 Introduction 2.1.1 Basic Definitions and Notation 2.1.2 Topological Basis and Subbasis 2.1.3 Nets 2.1.4 Continuous and Semicontinuous Functions 2.1.5 Open and Closed Maps: Homeomorphisms 2.1.6 Weak (or Initial) and Strong (or Final) Topologies 2.1.7 Compact Topological Spaces 2.1.8 Connectedness 2.1.9 Urysohn and Tietze Theorems 2.1.10 Paracompact and Baire Spaces 2.1.11 Polish and Souslin Sets 2.1.12 Michael Selection Theorem 2.1.13 The Space C(X;Y) 2.1.14 Elements of Algebraic Topology I: Homotopy 2.1.15 Elements of Algebraic Topology II: Homology 2.2 Problems 2.3 Solutions Bibliography 3 Measure, Integral and Martingales 3.1 Introduction 3.1.1 Basic Definitions and Notation 3.1.2 Measures and Outer Measures 3.1.3 The Lebesgue Measure 3.1.4 Atoms and Nonatomic Measures 3.1.5 Product Measures 3.1.6 Lebesgue-Stieltjes Measures 3.1.7 Measurable Functions 3.1.8 The Lebesgue Integral 3.1.9 Convergence Theorems 3.1.10 LP-Spaces 3.1.11 Multiple Integrals: Change of Variables 3.1.12 Uniform Integrability: Modes of Convergence 3.1.13 Signed Measures 3.1.14 Radon-Nikodym Theorem 3.1.15 Maximal Function and Lyapunov Convexity Theorem 3.1.16 Conditional Expectation and Martingales 3.2 Problems 3.3 Solutions Bibliography 4 Measures and Topology 4.1 Introduction 4.1.1 Borel and Baire a-Algebras 4.1.2 Regular and Radon Measures 4.1.3 Riesz Representation Theorem for Continuous Functions 4.1.4 Space of Probability Measures: Prohorov Theorem 4.1.5 Polish, Souslin and Borel Spaces 4.1.6 Measurable Multifunctions: Selection Theorems 4.1.7 Projection Theorems 4.1.8 Dual of LP(Ω) for 1 ≤ p ≤∞ 4.1.9 Sequences of Measures: Weak Convergence in LP(Ω) 4.1.10 Covering Theorems 4.1.11 Lebesgue Differentiation Theorem 4.1.12 Bounded Variation and Absolutely Continuous Functions 4.1.13 Hausdorff Measures: Change of Variables 4.1.14 Caratheodory Functions 4.2 Problems 4.3 Solutions Bibliography 5 Functional Analysis 5.1 Introduction 5.1.1 Locally Convex, Normed and Banach Spaces 5.1.2 Linear Operators: Quotient Spaces--Riesz Lemma 5.1.3 The Hahn-Banach Theorem 5.1.4 Adjoint Operators and Annihilators 5.1.5 The Three Basic Theorems of Linear Functional Analysis 5.1.6 The Weak Topology 5.1.7 The Weak* Topology 5.1.8 Reflexive Banach Spaces 5.1.9 Separable Banach Spaces 5.1.10 Uniformly Convex Spaces 5.1.11 Hilbert Spaces 5.1.12 Unbounded Linear Operators 5.1.13 Extremal Structure of Sets 5.1.14 Compact Operators 5.1.15 Spectral Theory 5.1.16 Differentiability and the Geometry of Banach Spaces 5.1.17 Best Approximation: Various Theorems for Banach Spaces 5.2 Problems 5.3 Solutions Bibliography Other Problem Books List of Symbols Index 編輯手記
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