泛函分析（英文版·原书第2版·典藏版） / 华章数学原版精品系列
定价：￥99.00
作者： [美]沃尔特·鲁丁
出版时间：2020-05-25
出版社：机械工业出版社
- 机械工业出版社
- 9787111654742
- 1-4
- 47229667-2
- 16开
- 2020-05-25
- 651
- 数学与应用数学
- 本科
作者简介
内容简介
本书不仅详细叙述了拓扑线性空间，包括若干子类局部凸空间、赋范空间、内积空间的公理系统、结构属性及其之上的强弱拓扑、共轭性，还深入论述了该学科离不开的几个专题，即形式上更为一般的三大基本定理与泛函延拓定理, Banach代数特别是Gelfand变换的基本理论，紧算子及其谱理论，自伴算子的谱理论，无界正常算子的谱理论以及Bonsall的闭值域定理，不变子空间的Lomonosov定理等；而且给出了以上基本理论的丰富多彩的应用，包括完整的关于广义函数、Fourier变换及其偏微分方程基本解的论述，对于Tauber型定理的应用，von Neumann的平均遍历定理，算子半群的Hille-Yosida定理并应用于发展方程等。
目录
Preface
About thd Author
Part I General Theory
Topological Vector Spaces
Introduction
Separation properties
Linear mappings
Finite-dimensional spaces
Metrization
Boundedness and continuity
Seminorms and local convexity
Quotient spaces
Examples
Exercises
2 Completeness
Baire category
The Banach-Steinhaus theorem
The open mapping theorem
The closed graph theorem
Bilinear mappings
Exercises
3 Convexity
The Hahn-Banach theorems
Weak topologies
Compact convex sets
Vector-valued integration
Holomorphic functions
Exercises
4 Duality in Banach Spaces
The normed dual of a normed space
Adjoints
Compact operators
Exercises
5 Some Applications
A continuity theorem
Closed subspaces of fi-spaces
The range of a vector-valued measure
A generalized Stone-Weierstrass theorem
Two interpolation theorems
Kakutani's fixed point theorem
Haar measure on compact groups
Uncomplemented subspaces
Sums of Poisson kernels
Two more fixed point theorems
Exercises
Part II Distributions and Fourier Transform
6 Test Functions and Distributions
Introduction
Test function spaces
Calculus with distributions
Localization
Supports of distributions
Distributions as derivatives
Convolutions
Exercises
7 Fourier Transforms
Basic properties
Tempered d]str]but]ons
Paley-Wiener theorems
Sobolev's lemma
Exercises
8 Applications to Differential Equations
Fundamental solutions
Elliptic equations
Exercises
9 Tauberian Theory
Wiener's theorem
The prime number theorem
The renewal equation
Exercises
Part III Banach Algebras and Spectral Theory
10 Banach Algebras
Introduction
Complex homomorphisms
Basic properties of spectra
Symbolic calculus
The group of invertible elements
Lomonosov's invariant subspace theorem
Exercises
11 Commutative Banach Algebras
Ideals and homomorphisms
Gelfand transforms
Involutions
Applications to noncommutative algebras
Positive functionals
Exercises
12 Bounded Operators on a Hilbert Space
Basic facts
Bounded operators
A commutativity theorem
Resolutions of the identity
The spectral theorem
Eigenvalues of normal operators
Positive operators and square roots
The group of invertible operators
A characterization of B*-algebras
An ergodic theorem
Exercises
13 Unbounded Operators
Introduction
Graphs and symmetric operators
The Cayley transform
Resolutions of the identity
The spectral theorem
Semigroups of operators
Exercises
Appendix A Compactness and Continuity
Appendix B Notes and Comments
Bibliography
List of Special Symbols
Index
About thd Author
Part I General Theory
Topological Vector Spaces
Introduction
Separation properties
Linear mappings
Finite-dimensional spaces
Metrization
Boundedness and continuity
Seminorms and local convexity
Quotient spaces
Examples
Exercises
2 Completeness
Baire category
The Banach-Steinhaus theorem
The open mapping theorem
The closed graph theorem
Bilinear mappings
Exercises
3 Convexity
The Hahn-Banach theorems
Weak topologies
Compact convex sets
Vector-valued integration
Holomorphic functions
Exercises
4 Duality in Banach Spaces
The normed dual of a normed space
Adjoints
Compact operators
Exercises
5 Some Applications
A continuity theorem
Closed subspaces of fi-spaces
The range of a vector-valued measure
A generalized Stone-Weierstrass theorem
Two interpolation theorems
Kakutani's fixed point theorem
Haar measure on compact groups
Uncomplemented subspaces
Sums of Poisson kernels
Two more fixed point theorems
Exercises
Part II Distributions and Fourier Transform
6 Test Functions and Distributions
Introduction
Test function spaces
Calculus with distributions
Localization
Supports of distributions
Distributions as derivatives
Convolutions
Exercises
7 Fourier Transforms
Basic properties
Tempered d]str]but]ons
Paley-Wiener theorems
Sobolev's lemma
Exercises
8 Applications to Differential Equations
Fundamental solutions
Elliptic equations
Exercises
9 Tauberian Theory
Wiener's theorem
The prime number theorem
The renewal equation
Exercises
Part III Banach Algebras and Spectral Theory
10 Banach Algebras
Introduction
Complex homomorphisms
Basic properties of spectra
Symbolic calculus
The group of invertible elements
Lomonosov's invariant subspace theorem
Exercises
11 Commutative Banach Algebras
Ideals and homomorphisms
Gelfand transforms
Involutions
Applications to noncommutative algebras
Positive functionals
Exercises
12 Bounded Operators on a Hilbert Space
Basic facts
Bounded operators
A commutativity theorem
Resolutions of the identity
The spectral theorem
Eigenvalues of normal operators
Positive operators and square roots
The group of invertible operators
A characterization of B*-algebras
An ergodic theorem
Exercises
13 Unbounded Operators
Introduction
Graphs and symmetric operators
The Cayley transform
Resolutions of the identity
The spectral theorem
Semigroups of operators
Exercises
Appendix A Compactness and Continuity
Appendix B Notes and Comments
Bibliography
List of Special Symbols
Index
