Book:Walter Rudin/Functional Analysis/Second Edition

Walter Rudin: Functional Analysis (2nd Edition)

Published $\text {1991}$, McGraw-Hill

ISBN 978-0070619883

Subject Matter

Functional Analysis

Contents

Preface

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 functions 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 functions

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 Homomorhpisms

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

Further Editions

• 1973: Walter Rudin: Functional Analysis