简介
A large mathematical community throughout the world actively works in functional analysis and uses profound techniques from topology.聽As the first monograph to approach the topic of topological vector spaces from the perspective of descriptive topology, this work provides also new insights into the connections between the topological properties of linear function spaces and their role in functional analysis.聽 Descriptive Topology in Selected Topics of Functional Analysis is a self-contained volume that applies recent developments and classical results in descriptive topology to study the classes of infinite-dimensional topological vector spaces that appear in functional analysis. Such spaces include Fr茅chet spaces, LF-spaces and their duals, and聽the space of continuous real-valued functions C(X) on a completely regular Hausdorff space X, to name a few. These vector spaces appear in distribution theory, differential equations, complex analysis, and various other areas of functional analysis. Written by three experts in the field, this book is a treasure trove for researchers and graduate students studying the interplay among the areas of point-set and descriptive topology, modern analysis, set theory, topological vector spaces and Banach spaces, and continuous function spaces.聽聽Moreover, it will serve as a reference for present and future work done in this area and could serve as a valuable supplement to advanced graduate courses in functional analysis, set-theoretic topology, or the theory of function spaces.
目录
Descriptive Topology in Selected Topics of Functional Analysis 2
Preface 5
Contents 6
Chapter 1: Overview 10
1.1 General comments and historical facts 16
Chapter 2: Elementary Facts about Baire and Baire-Type Spaces 21
2.1 Baire spaces and Polish spaces 21
2.2 A characterization of Baire topological vector spaces 26
2.3 Arias de Reyna-Valdivia-Saxon theorem 28
2.4 Locally convex spaces with some Baire-type conditions 32
2.5 Strongly realcompact spaces X and spaces Cc(X) 44
2.6 Pseudocompact spaces, Warner boundedness and spaces Cc(X) 54
2.7 Sequential conditions for locally convex Baire-type spaces 64
Chapter 3: K-analytic and Quasi-Suslin Spaces 71
3.1 Elementary facts 71
3.2 Resolutions and K-analyticity 79
3.3 Quasi-(LB)-spaces 90
3.4 Suslin schemes 99
3.5 Applications of Suslin schemes to separable metrizable spaces 101
3.6 Calbrix-Hurewicz theorem 109
Chapter 4: Web-Compact Spaces and Angelic Theorems 117
4.1 Angelic lemma and angelicity 117
4.2 Orihuela's angelic theorem 119
4.3 Web-compact spaces 121
4.4 Subspaces of web-compact spaces 124
4.5 Angelic duals of spaces C(X) 126
4.6 About compactness via distances to function spaces C(K) 128
Chapter 5: Strongly Web-Compact Spaces and a Closed Graph Theorem 144
5.1 Strongly web-compact spaces 144
5.2 Products of strongly web-compact spaces 145
5.3 A closed graph theorem for strongly web-compact spaces 147
Chapter 6: Weakly Analytic Spaces 150
6.1 A few facts about analytic spaces 150
6.2 Christensen's theorem 156
6.3 Subspaces of analytic spaces 162
6.4 Trans-separable topological spaces 164
6.5 Weakly analytic spaces need not be analytic 171
6.6 More about analytic locally convex spaces 174
6.7 Weakly compact density condition 175
6.8 More examples of nonseparable weakly analytic tvs 181
Chapter 7: K-analytic Baire Spaces 189
7.1 Baire tvs with a bounded resolution 189
7.2 Continuous maps on spaces with resolutions 193
Chapter 8: A Three-Space Property for Analytic Spaces 198
8.1 An example of Corson 198
8.2 A positive result and a counterexample 201
Chapter 9: K-analytic and Analytic Spaces Cp(X) 205
9.1 A theorem of Talagrand for spaces Cp(X) 205
9.2 Theorems of Christensen and Calbrix for Cp(X) 208
9.3 Bounded resolutions for Cp(X) 219
9.4 Some examples of K-analytic spaces Cp(X) and Cp(X,E) 234
9.5 K-analytic spaces Cp(X) over a locally compact group X 235
9.6 K-analytic group Xp of homomorphisms 238
Chapter 10: Precompact Sets in (LM)-Spaces and Dual Metric Spaces 242
10.1 The case of (LM)-spaces: elementary approach 242
10.2 The case of dual metric spaces: elementary approach 244
Chapter 11: Metrizability of Compact Sets in the Class G 246
11.1 The class G: examples 246
11.2 Cascales-Orihuela theorem and applications 248
Chapter 12: Weakly Realcompact Locally Convex Spaces 254
12.1 Tightness and quasi-Suslin weak duals 254
12.2 A Kaplansky-type theorem about tightness 257
12.3 K-analytic spaces in the class G 261
12.4 Every WCG Fr茅chet space is weakly K-analytic 263
12.5 Amir-Lindenstrauss theorem 269
12.6 An example of Pol 274
12.7 More about Banach spaces C(X) over compact scattered X 279
Chapter 13: Corson's Property (C) and Tightness 282
13.1 Property (C) and weakly Lindel枚f Banach spaces 282
13.2 The property (C) for Banach spaces C(X) 287
Chapter 14: Fr茅chet-Urysohn Spaces and Groups 291
14.1 Fr茅chet-Urysohn topological spaces 291
14.2 A few facts about Fr茅chet-Urysohn topological groups 293
14.3 Sequentially complete Fr茅chet-Urysohn spaces are Baire 298
14.4 Three-space property for Fr茅chet-Urysohn spaces 301
14.5 Topological vector spaces with bounded tightness 304
Chapter 15: Sequential Properties in the Class G 306
15.1 Fr茅chet-Urysohn spaces are metrizable in the class G 306
15.2 Sequential (LM)-spaces and the dual metric spaces 312
15.3 (LF)-spaces with the property C3- 321
Chapter 16: Tightness and Distinguished Fr茅chet Spaces 328
16.1 A characterization of distinguished spaces 328
16.2 G-bases and tightness 335
16.3 G-bases, bounding, dominating cardinals, and tightness 339
16.4 More about the Wulbert-Morris space Cc(omega1) 350
Chapter 17: Banach Spaces with Many Projections 356
17.1 Preliminaries, model-theoretic tools 356
17.2 Projections from elementary submodels 362
17.3 Lindel枚f property of weak topologies 365
17.4 Separable complementation property 366
17.5 Projectional skeletons 370
17.6 Norming subspaces induced by a projectional skeleton 376
17.7 Sigma-products 381
17.8 Markushevich bases, Plichko spaces and Plichko pairs 384
17.9 Preservation of Plichko spaces 389
Chapter 18: Spaces of Continuous Functions over Compact Lines 396
18.1 General facts 396
18.2 Nakhmanson's theorem 399
18.3 Separable complementation 400
Chapter 19: Compact Spaces Generated by Retractions 406
19.1 Retractive inverse systems 406
19.2 Monolithic sets 410
19.3 Classes R and RC 412
19.4 Stability 413
19.5 Some examples 416
19.6 The first cohomology functor 419
19.7 Compact lines 423
19.8 Valdivia and Corson compact spaces 426
19.9 Preservation theorem 433
19.10 Retractional skeletons 435
19.11 Primarily Lindel枚f spaces 439
19.12 Corson compact spaces and WLD spaces 441
19.13 A dichotomy 443
19.14 Alexandrov duplications 447
19.15 Valdivia compact groups 449
19.16 Compact lines in class R 452
19.17 More on Eberlein compact spaces 457
Chapter 20: Complementably Universal Banach Spaces 467
20.1 Amalgamation lemma 467
20.2 Embedding-projection pairs 469
20.3 A complementably universal Banach space 471
References 475
Index 491
Preface 5
Contents 6
Chapter 1: Overview 10
1.1 General comments and historical facts 16
Chapter 2: Elementary Facts about Baire and Baire-Type Spaces 21
2.1 Baire spaces and Polish spaces 21
2.2 A characterization of Baire topological vector spaces 26
2.3 Arias de Reyna-Valdivia-Saxon theorem 28
2.4 Locally convex spaces with some Baire-type conditions 32
2.5 Strongly realcompact spaces X and spaces Cc(X) 44
2.6 Pseudocompact spaces, Warner boundedness and spaces Cc(X) 54
2.7 Sequential conditions for locally convex Baire-type spaces 64
Chapter 3: K-analytic and Quasi-Suslin Spaces 71
3.1 Elementary facts 71
3.2 Resolutions and K-analyticity 79
3.3 Quasi-(LB)-spaces 90
3.4 Suslin schemes 99
3.5 Applications of Suslin schemes to separable metrizable spaces 101
3.6 Calbrix-Hurewicz theorem 109
Chapter 4: Web-Compact Spaces and Angelic Theorems 117
4.1 Angelic lemma and angelicity 117
4.2 Orihuela's angelic theorem 119
4.3 Web-compact spaces 121
4.4 Subspaces of web-compact spaces 124
4.5 Angelic duals of spaces C(X) 126
4.6 About compactness via distances to function spaces C(K) 128
Chapter 5: Strongly Web-Compact Spaces and a Closed Graph Theorem 144
5.1 Strongly web-compact spaces 144
5.2 Products of strongly web-compact spaces 145
5.3 A closed graph theorem for strongly web-compact spaces 147
Chapter 6: Weakly Analytic Spaces 150
6.1 A few facts about analytic spaces 150
6.2 Christensen's theorem 156
6.3 Subspaces of analytic spaces 162
6.4 Trans-separable topological spaces 164
6.5 Weakly analytic spaces need not be analytic 171
6.6 More about analytic locally convex spaces 174
6.7 Weakly compact density condition 175
6.8 More examples of nonseparable weakly analytic tvs 181
Chapter 7: K-analytic Baire Spaces 189
7.1 Baire tvs with a bounded resolution 189
7.2 Continuous maps on spaces with resolutions 193
Chapter 8: A Three-Space Property for Analytic Spaces 198
8.1 An example of Corson 198
8.2 A positive result and a counterexample 201
Chapter 9: K-analytic and Analytic Spaces Cp(X) 205
9.1 A theorem of Talagrand for spaces Cp(X) 205
9.2 Theorems of Christensen and Calbrix for Cp(X) 208
9.3 Bounded resolutions for Cp(X) 219
9.4 Some examples of K-analytic spaces Cp(X) and Cp(X,E) 234
9.5 K-analytic spaces Cp(X) over a locally compact group X 235
9.6 K-analytic group Xp of homomorphisms 238
Chapter 10: Precompact Sets in (LM)-Spaces and Dual Metric Spaces 242
10.1 The case of (LM)-spaces: elementary approach 242
10.2 The case of dual metric spaces: elementary approach 244
Chapter 11: Metrizability of Compact Sets in the Class G 246
11.1 The class G: examples 246
11.2 Cascales-Orihuela theorem and applications 248
Chapter 12: Weakly Realcompact Locally Convex Spaces 254
12.1 Tightness and quasi-Suslin weak duals 254
12.2 A Kaplansky-type theorem about tightness 257
12.3 K-analytic spaces in the class G 261
12.4 Every WCG Fr茅chet space is weakly K-analytic 263
12.5 Amir-Lindenstrauss theorem 269
12.6 An example of Pol 274
12.7 More about Banach spaces C(X) over compact scattered X 279
Chapter 13: Corson's Property (C) and Tightness 282
13.1 Property (C) and weakly Lindel枚f Banach spaces 282
13.2 The property (C) for Banach spaces C(X) 287
Chapter 14: Fr茅chet-Urysohn Spaces and Groups 291
14.1 Fr茅chet-Urysohn topological spaces 291
14.2 A few facts about Fr茅chet-Urysohn topological groups 293
14.3 Sequentially complete Fr茅chet-Urysohn spaces are Baire 298
14.4 Three-space property for Fr茅chet-Urysohn spaces 301
14.5 Topological vector spaces with bounded tightness 304
Chapter 15: Sequential Properties in the Class G 306
15.1 Fr茅chet-Urysohn spaces are metrizable in the class G 306
15.2 Sequential (LM)-spaces and the dual metric spaces 312
15.3 (LF)-spaces with the property C3- 321
Chapter 16: Tightness and Distinguished Fr茅chet Spaces 328
16.1 A characterization of distinguished spaces 328
16.2 G-bases and tightness 335
16.3 G-bases, bounding, dominating cardinals, and tightness 339
16.4 More about the Wulbert-Morris space Cc(omega1) 350
Chapter 17: Banach Spaces with Many Projections 356
17.1 Preliminaries, model-theoretic tools 356
17.2 Projections from elementary submodels 362
17.3 Lindel枚f property of weak topologies 365
17.4 Separable complementation property 366
17.5 Projectional skeletons 370
17.6 Norming subspaces induced by a projectional skeleton 376
17.7 Sigma-products 381
17.8 Markushevich bases, Plichko spaces and Plichko pairs 384
17.9 Preservation of Plichko spaces 389
Chapter 18: Spaces of Continuous Functions over Compact Lines 396
18.1 General facts 396
18.2 Nakhmanson's theorem 399
18.3 Separable complementation 400
Chapter 19: Compact Spaces Generated by Retractions 406
19.1 Retractive inverse systems 406
19.2 Monolithic sets 410
19.3 Classes R and RC 412
19.4 Stability 413
19.5 Some examples 416
19.6 The first cohomology functor 419
19.7 Compact lines 423
19.8 Valdivia and Corson compact spaces 426
19.9 Preservation theorem 433
19.10 Retractional skeletons 435
19.11 Primarily Lindel枚f spaces 439
19.12 Corson compact spaces and WLD spaces 441
19.13 A dichotomy 443
19.14 Alexandrov duplications 447
19.15 Valdivia compact groups 449
19.16 Compact lines in class R 452
19.17 More on Eberlein compact spaces 457
Chapter 20: Complementably Universal Banach Spaces 467
20.1 Amalgamation lemma 467
20.2 Embedding-projection pairs 469
20.3 A complementably universal Banach space 471
References 475
Index 491
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