分析学练习(第1部分)

本书共分为五部分，第一部分介绍了度量空间的相关内容；第二部分介绍了拓扑空间的相关内容，包含一些代数拓扑的介绍性资料；第三部分介绍了测度的相关理论，包含整合与鞅；第四部分介绍了测度与拓扑，其中涉及测度理论与拓扑之间相互影响的内容；第五部分介绍了泛函分析，重点强调了巴拿赫空间理论，每部分主要介绍了基本理论，不仅给出了相关的定义和结果，还阐述了有关概念和结果的注释和评论，可以帮助读者在解决问题之前掌握其理论知识。本书适合大学师生及数学爱好者参考阅读。

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
编辑手记