作者：(波)莱谢克·加林斯基(Leszek G
页数：1047页
出版社：哈尔滨工业大学出版社
出版日期：2021
ISBN：9787560392288
电子书格式：pdf/epub/txt
内容简介
本书是一部版权引进自著名出版公司斯普林格出版公司的英文原版数学著作。这本书的目的是回顾分析学中的基本理论及其问题与解决方法.通过这些问题,读者可以检验自己对这些理论的理解程度,也可以发现这些理论的延伸和文献中不规范的附加结果.本书的主题或多或少涵盖了标准的本科高年级和研究生的分析课程的一些内容。
目录
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最 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
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最 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
