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Functional Analysis

Edition: 
2
Publisher: 
McGraw-Hill
Number of Pages: 
448
Price: 
123.75
ISBN: 
0070542368
Date Received: 
Monday, July 10, 2006
Reviewable: 
No
Include In BLL Rating: 
Yes
Reviewer Email Address: 
Walter Rudin
Publication Date: 
1991
Format: 
Hardcover
Category: 
Monograph

 

Preface.

PART ONE: GENERAL THEORY

1. Topological Vector Space

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 TWO: 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 THREE: 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 Hillbert 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


Publish Book: 
Modify Date: 
Monday, July 10, 2006

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