Topological Uniform Structures

Cover

S Title

TOPOLOGICAL UNIFORM STRUCTURES

Copyright © 1978, 1988 by Warren Page
ISBN 0-486-65808-2
[QA61 l.25.P33 1988] 514'.3202-dcl9

Dedication

Preface

A Guide to this Book

Contents

Symbols and Notations

CHAPTER I Uniform Spaces
1. Entourage Uniformities
Uniform Type Continuity
PROBLEMS
References for Further Study
2. Covering Uniformities
PROBLEMS
3. Projective and Inductive Limits
Projective Limits
Inductive Limits
PROBLEMS
4. Uniformities and Gages
Gage Spaces
Uniformly Continuous Pseudometrics
PROBLEMS
References for Further Study
5. Total Boundedness
PROBLEMS
6. Completeness
Uniform Space Completions
Completions and Compactifications
PROBLEMS
References for Further Study
7. Function Spaces
S-Convergence
Equicontinuity and Uniform Equicontinuity
PROBLEMS
References for Further Study

CHAPTER II Topological Groups
8. Topological Groups are Uniformizible
Completeness for Topological Groups
Connectedness Considerations
REFERENCES FOR FURTHER STUDY
PROBLEMS
9. Projective and Inductive Limits
Inductive Limits
Topological Direct Products
PROBLEMS
10. Open Mapping and Closed Graph Theorems
PROBLEM
11. Unitary Representations and Character Groups
Unitary Representations
Character Groups
Concluding Remarks
PROBLEMS
References for further Study
12. Haar Measure and Integrationt
Restricted and Extended Haar Concepts
Computation and Examples of Haar Concepts
PROBLEMS
References for Further Study

CHAPTER III Topological Vector Spaces
13. TVSps and Topological Groups
Boundedness in a TVS
S-Topologies on BL(X, Y)
PROBLEMS
References for Further Study
14. Locally Convex TVSps
PROBLEMS
References for Further Study
15. Projective and Inductive Limits
Projective Limits
Inductive Limits
PROBLEMS
References for Further Study
16. Vector-Valued Measure TVSps
PROBLEMS
References for Further Study
17. Hahn Banach Theorems
Krein Milman Theorem
PROBLEMS
References for Future Study
18. Duality Theory
Polar Sets
Transpose of a Linear Mapping
Polar Topologies
Grothendieck's Completion Theorem
PROBLEMS
References for Further Study
19. Bornological and Barreled Spaces
Barreled Spaces
PROBLEMS
References for Further Study
20. Reflexive and Montel Spaces
Semireflexive and Reflexive Spaces
Semi-Montel and Montel Spaces
PROBLEMS
References for Further Study
21. Full Completeness: Open Mapping and Closed Graph Theorems
PROBLEMS
References for Further Study

CHAPTER IV Topological Algebras
22. Algebraic Preliminaries
PROBLEMS
23. Normed and Normed *-Algebras
*-Algebras
PROBLEMS
24. TV As and LMC Algebras
Topological Vector Algebras
LMC Algebras
PROBLEMS
25. Q-Algebras
PROBLEMS
References for Further Study
26. Complete, Complex, LMCT2 Q-Algebras
A and A*-Algebrast
Completely Regular and Normal Algebras
Hull Kernel Topology
The Silov Boundary
PROBLEMS
References for Further Study

CHAPTER V Abstract Harmonic Analysis
27. The Algebra L1(G)
Fourier Transforms
For the remainder of this section G is assumed to be abelian.
PROBLEMS
28. The Algebra M ( G) and Its Components
Fourier Stieltjes Transform
For the remainder of this section, G is assumed abellan
PROBLEMS
References for Further Study
29. Fourier Analysis on LCT 2A Groups
Bochner's Theorem
The Inversion Theorem
Fourier Transforms on L2( G)
Pontryagin Duality Theorem
PROBLEMS
References for Further Study

APPENDIX T Topology
t.1. Neighborhood Systems
t.2. Closure Axioms
t.3. Category Notions
t.4. Almost Open Sets. (Cech [ 16])
t.S. Coverings
t.6. Nets and Filters
t.7. Compactness and Compactifications
t.8. Function Spaces
t.9. Stone Weierstrass Theorems

APPENDIX M Measure and Integration
m.1. Measurable Spaces and Measure Spaces
m.2. Measurable Functions
m.3. Integrable Functions
m.4. Product Measure and Fubini's Theorem
m.5. Signed Measures
m.6. Complex Measures
m.7. Absolute Continuity and Mutual Singularity
m.8. Measure and Integration in Locally Compact, T2 Spaces

APPENDIX A Linear Algebra
a.1. Orthogonality
a.2. Completeness
a.3. Sesquilinear Functionals

Bibliography
INDEX
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