泛函分析教程  第2版

泛函分析教程 第2版

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内容简介

本书作者擅长写教科书,以选材仔细、论述清晰、实例丰富著称。本书是一部代理科研究生使用的泛函分析教材,读者只需具备积分和测度论的知识即可阅读。全书充分体现了作者的著书风格,以实例先行,从具体到一般,从浅入深,并配有许多精心挑选的例题和习题。

作者简介

J. B. 康威(J. B. Conway),美国田纳西大学(Tennessee University)数学系教授,本书和《单变量函数》(2卷集)被广泛用于研究生教材。

章节目录

Preface

Preface to the Second Edition

CHAPTER I

Hilbert Spaces

1.Elementary Properties and Examples

2.Orthogonality

3.The Riesz Representation Theorem

4.Orthonormal Sets of Vectors and Bases

5.Isomorphic Hilbert Spaces and the Fourier Transform for the Circle

6.The Direct Sum of Hilbert Spaces

CHAPTER II

Operators on Hilbert Space

1.Elementary Properties and Examples

2.The Adjoint of an Operator

3.Projections and Idempotents;Invariant and Reducing Subspaces

4.Compact Operators

5.The Diagonalization of Compact Self-Adjoint Operators

6.An Application:Sturm-Liouville Systems

7.The Spectral Theorem and Functional Calculus for Compact Normai

Operators

8.Unitary Equivalence for Compact Normai Operators

CHAPTER III

Banach Spaces

1.Elementary Properties and Examples

2.Linear Operators on Normed Spaces

3.Finite Dimensional Normed Spaces

4.Quotients and Products of Normed Spaces

5.Linear Functionals

6.The Hahn-Banach Theorem

7.An Application:Banach Limits

8.An Application:Runge's Theorem

9.An Application:Ordered Vector Spaces

10.The Dual of a Quotient Space and a Subspace

11.Reflexive Spaces

12.The Open Mapping and Closed Graph Theorems

13.Complemented Subspaces of a Banach Space

14.The Principle of Uniform Boundedness

CHAPTER IV

Locally Convex Spaces

S1.Elementary Properties and Examples

2.Metrizable and Normable Locally Convex Spaces

3.Some Geometric Consequences of the Hahn-Banach Theorem

4.Some Examples of the Dual Space of a Locally Convex Space

5.Inductive Limits and the Space of Distributions

CHAPTER V

Weak Topologies

1.Duality

2.The Dual of a Subspace and a Quotient Space

3.Alaoglu's Theorem

84.Reflexivity Revisited

5.Separability and Metrizability

S6.An Application:The Stone-Cech Compactification

87.The Krein-Milman Theorem

8.An Application:The Stone-Weierstrass Theorem

9.The Schauder Fixed Point Theorem

10.The Ryll-Nardzewski Fixed Point Theorem

11.An Application:Haar Measure on a Compact Group

12.The Krein-Smulian Theorem

13.Weak Compactness

CHAPTER VI

Linear Operators on a Banach Space

1.The Adjoint of a Linear Operator

2.The Banach-Stone Theorem

3.Compact Operators

4.Invariant Subspaces

5.Weakly Compact Operators

CHAPTER VII

Banach Algebras and Spectral Theory for

Operators on a Banach Space

1.Elementary Properties and Examples

2.Ideals and Quotients

3.The Spectrum

4.The Riesz Functional Calculus

5.Dependence of the Spectrum on the Aigebra

6.The Spectrum of a Linear Operator

7.The Spectral Theory of a Compact Operator

8.Abelian Banach Algebras

9.The Group Algebra of a Locally Compact Abelian Group

CHAPTER VIII

C-Algebras

1.Elementary Properties and Examples

2.Abelian C-Algebras and the Functional Calculus in C-Algebras

3.The Positive Elements in a C-Algebra

4.Ideals and Quotients of C-Algebras

5.Representations of C-Algebras and the Gelfand-Naimark-Segal

Construction

CHAPTER IX

Normal Operators on Hilbert Space

1.Spectral Measures and Representations of Abelian C-Algebras

2.The Spectral Theorem

3.Star-Cyclic Normal Operators

4.Some Applications of the Spectral Theorem

5.Topologies on (X)

6.Commuting Operators

7.Abelian von Neumann Algebras

8.The Functional Calculus for Normal Operators:

The Conclusion of the Saga

Invariant Subspaces for Normal Operators

9.Multiplicity Theory for Normal Operators:

10.A Complete Set of Unitary Invariants

CHAPTER X

Unbounded Operators

1.Basic Properties and Examples

2.Symmetric and Self-Adjoint Operators

3.The Cayley Transform

4.Unbounded Normal Operators and the Spectral Theorem

S5.Stone's Theorem

6.The Fourier Transform and Differentiation

7.Moments6

CHAPTER XI

Fredholm Theory

1.The Spectrum Revisited

2.Fredholm Operators

3.The Fredholm Index

4.The Essential Spectrum

5.The Components of Sg

6.A Finer Analysis of the Spectrum

APPENDIX A

Preliminaries

1.Linear Algebra

2.Topology

APENDIX B

The Dual of LP(u)

APPENDIX C

The Dual of Co(X)

Bibliography

List of Symbols

Index

泛函分析教程 第2版是2019年由世界图书出版公司出版,作者[美]J.B.康威。

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