泛涵分析
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作者沃尔特·鲁丁
出版社机械工业出版社
ISBN9787111654742
出版时间2020-05
装帧平装
开本16开
定价99元
货号1202071236
上书时间2024-06-05
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作者简介
沃尔特·鲁丁（Walter Rudin），1953年于杜克大学获得数学博士学位。曾先后执教于麻省理工学院、罗切斯特大学、威斯康星大学麦迪逊分校、耶鲁大学等。他的主要研究兴趣集中在调和分析和复变函数上。除本书外，他还著有《Functional Analysis》（泛函分析）和《Principles of Mathematical Analysis》（数学分析原理）等其他名著。这些教材已被翻译成十几种语言，在世界各地广泛使用。
目录
Preface
About the Author
Part I General Theory
1 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 Lp-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 Transforms
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 distributions
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
内容摘要
本书是国际著名教材，在材料的取舍和处理手法上很有特色，对某些公理进行了准确描述，并精彩地讨论了一些深入的专题，还介绍了在其他数学分支（如微分方程）中有价值的应用。用作者自己的话来讲，他并不期望写一部百科全书，而是为进一步的探索打开通道。
本书叙述清楚，论证严谨，不少地方的注释相当精辟并具有启发性，可作为高等院校数学专业高年级本科生和研究生的教材和参考书。
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