Convergence Structures and Applications to Functional Analysis / Edition 1

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This text offers a rigorous introduction into the theory and methods of convergence spaces and gives concrete applications to the problems of functional analysis. While there are a few books dealing with convergence spaces and a great many on functional analysis, there are none with this particular focus.

The book demonstrates the applicability of convergence structures to functional analysis. Highlighted here is the role of continuous convergence, a convergence structure particularly appropriate to function spaces. It is shown to provide an excellent dual structure for both topological groups and topological vector spaces.

Readers will find the text rich in examples. Of interest, as well, are the many filter and ultrafilter proofs which often provide a fresh perspective on a well-known result.

Audience: This text will be of interest to researchers in functional analysis, analysis and topology as well as anyone already working with convergence spaces. It is appropriate for senior undergraduate or graduate level students with some background in analysis and topology.

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Product Details

  • ISBN-13: 9781402005664
  • Publisher: Springer Netherlands
  • Publication date: 3/31/2002
  • Edition description: 2002
  • Edition number: 1
  • Pages: 264
  • Product dimensions: 9.21 (w) x 6.14 (h) x 0.69 (d)

Table of Contents

1 Convergence spaces 1
1.1 Preliminaries 1
1.2 Initial and final convergence structures 4
1.3 Special convergence spaces, modifications 10
1.4 Compactness 21
1.5 The continuous convergence structure 25
1.6 Countability properties and sequences in convergence spaces 42
1.7 Sequential convergence structures 51
1.8 Categorical aspects 57
2 Uniform convergence spaces 59
2.1 Generalities on uniform convergence spaces 59
2.2 Initial and final uniform convergence structures 67
2.3 Complete uniform convergence spaces 69
2.4 The Arzela-Ascoli theorem 71
2.5 The uniform convergence structure of a convergence group 75
3 Convergence vector spaces 79
3.1 Convergence groups 79
3.2 Generalities on convergence vector spaces 85
3.3 Initial and final vector space convergence structures 88
3.4 Projective and inductive limits of convergence vector spaces 96
3.5 The locally convex topological modification 102
3.6 Countability axioms for convergence vector spaces 109
3.7 Boundedness 111
3.8 Notes on bornological vector spaces 116
4 Duality 119
4.1 The dual of a convergence vector space 119
4.2 Reflexivity 125
4.3 The dual of a locally convex topological vector space 131
4.4 An application of continuous duality 148
4.5 Notes 152
5 Hahn-Banach extension theorems 153
5.1 General results 154
5.2 Hahn-Banach spaces 160
5.3 Extending to the adherence 164
5.4 Strong Hahn-Banach spaces 172
5.5 An application to partial differential equations 178
5.6 Notes 181
6 The closed graph theorem 183
6.1 Ultracompleteness 184
6.2 The main theorems 187
6.3 An application to web spaces 192
7 The Banach-Steinhaus theorem 195
7.1 Equicontinuous sets 196
7.2 Banach-Steinhaus pairs 198
7.3 The continuity of bilinear mappings 204
8 Duality theory for convergence groups 207
8.1 Reflexivity 208
8.2 Duality for convergence vector spaces 215
8.3 Subgroups and quotient groups 217
8.4 Topological groups 224
8.5 Groups of unimodular continuous functions 232
8.6 c- and co-duality for topological groups 240
Bibliography 247
List of Notations 257
Index 259
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