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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