Locally convex spaces

Contents......Page 9
1.1 Generalities......Page 15
1.2 Elementary Constructions......Page 16
1.3 Linear Maps......Page 17
1.4 Linear Independence......Page 19
1.5 Linear Forms......Page 20
1.6 Bilinear Maps and Tensor Products......Page 22
1.7 Some Examples......Page 25
2.1 Generalities......Page 30
2.2 Circled and Absorbent Sets......Page 32
2.3 Bounded Sets. Continuous Linear Forms......Page 34
2.4 Projective Topologies......Page 35
2.5 A Universal Characterization of Products......Page 36
2.6 Projective Limits......Page 37
2.7 F-Seminorms......Page 38
2.8 Metrizable Tvs......Page 40
2.9 Projective Representation of Tvs......Page 42
2.10 Linear Topologies on Function and Sequence Spaces......Page 43
2.11 References......Page 54
3.1 Some General Concepts......Page 56
3.2 Some Completeness Concepts......Page 57
3.3 Completion of a Tvs......Page 59
3.4 Extension of Uniformly Continuous Maps......Page 61
3.5 Precompact Sets......Page 64
3.6 Examples......Page 66
3.7 References......Page 73
4.1 Generalities......Page 74
4.2 Quotients of Tvs......Page 76
4.3 Direct Sums......Page 78
4.4 Some Completeness Results......Page 80
4.5 Inductive Limits......Page 82
4.6 Strict Inductive Limits......Page 84
4.7 References......Page 86
5.1 Baire Category......Page 87
5.2 Webs in Tvs......Page 89
5.3 Stability Properties of Webbed Tvs......Page 90
5.4 The Closed Graph Theorem......Page 92
5.5 Some Consequences......Page 95
5.6 Strictly Webbed Tvs......Page 96
5.7 Some Examples......Page 97
5.8 References......Page 99
6.1 r-Convex Sets......Page 101
6.2 r-Convex Sets in Tvs......Page 103
6.3 Gauge Functionals and r-Seminorms......Page 104
6.4 Continuity Properties of Gauge Functionals......Page 106
6.5 Definition and Basic Properties of Lc,s......Page 108
6.6 Some Permanence Properties of Lc,s......Page 109
6.7 Bounded, Precompact, and Compact Sets......Page 112
6.8 Locally Bounded Tvs......Page 114
6.9 Linear Mappings Between r-Normable Tvs......Page 117
6.10 Examples......Page 119
6.11 References......Page 124
7.1 Sublinear Functionals......Page 125
7.2 Extension Theorem for Lcs......Page 127
7.3 Separation Theorems......Page 130
7.4 Extension Theorems for Normed Spaces......Page 132
7.5 The Krein-Milman Theorem......Page 133
7.6 The Riesz Representation Theorem......Page 137
7.7 References......Page 144
8.1 Dual Pairings and Weak Topologies......Page 145
8.2 Polarization......Page 148
8.3 Barrels and Disks......Page 150
8.4 Bornologies and B-Topologies......Page 152
8.5 Equicontinuous Sets and Compactologies......Page 156
8.6 Continuity of Linear Maps......Page 160
8.7 Duality of Subspaces and Quotients......Page 163
8.8 Duality of Products and Direct Sums......Page 165
8.9 The Stone-Weierstrass Theorem......Page 169
8.10 References......Page 173
9.1 Continuous Convergence......Page 174
9.2 Grothendieck's Completeness Theorem......Page 176
9.3 The Topologies \gamma and \gamma^t......Page 178
9.4 The Banach-Dieudonne Theorem......Page 181
9.5 B-Completeness and Related Properties......Page 183
9.6 Open and Nearly Open Mappings......Page 184
9.7 Application to B-Completeness......Page 186
9.8 On Weak Compactness......Page 189
9.9 References......Page 193
10.1 B-Convergence. Local Convergence......Page 195
10.2 Local Completeness......Page 197
10.3 Equicontinuous Convergence. The Topologies \eta^t and \eta......Page 199
10.4 Schwartz Topologies......Page 201
10.5 A Universal Schwartz Space......Page 204
10.6 Diametral Dimension. Power Series Spaces......Page 207
10.7 Quasi-Normable Lcs......Page 214
10.8 Application to Continuous Function Spaces......Page 216
10.9 References......Page 217
11.1 Barrelled Lcs......Page 219
11.2 Quasi-Barrelled Lcs......Page 222
11.3 Some Permanence Properties......Page 223
11.4 Semi-Reflexive and Reflexive Lcs......Page 227
11.5 Semi-Montel and Montel Spaces......Page 229
11.6 On Fréchet-Montel Spaces......Page 231
11.7 Application to Continuous Function Spaces......Page 233
11.8 On Uniformly Convex Banach Spaces......Page 236
11.9 On Hilbert Spaces......Page 241
11.10 References......Page 247
12.1 \l_\infty-Barrelled and c_0-Barrelled Lcs......Page 249
12.2 \aleph_0-Barrelled Lcs......Page 251
12.3 Absorbent and Bornivorous Sequences......Page 253
12.4 DF-Spaces, gDF-Spaces, and df-Spaces......Page 257
12.5 Relations to Schwartz Topologies......Page 263
12.6 Application to Continuous Function Spaces......Page 266
12.7 References......Page 269
13.1 Generalities......Page 271
13.2 B-Convergent and Rapidly B-Convergent Sequences......Page 273
13.3 Associated Bornological and Ultrabornological Spaces......Page 276
13.4 On the Topology \beta(E', E)^{bor}......Page 279
13.5 Permanence Properties......Page 281
13.6 Application to Continuous Function Spaces......Page 283
13.7 References......Page 288
14.1 Biorthogonal Sequences......Page 289
14.2 Bases and Schauder Bases......Page 292
14.3 Weak Bases. Equicontinuous Bases......Page 295
14.4 Examples and Additional Remarks......Page 299
14.5 Shrinking and Boundedly Complete Bases......Page 302
14.6 On Summable Sequences......Page 305
14.7 Unconditional and Absolute Bases......Page 309
14.8 Orthonormal Bases in Hilbert Spaces......Page 315
14.9 References......Page 320
15.1 Generalities on Projective Tensor Products......Page 323
15.2 Tensor Product and Linear Mappings......Page 326
15.3 Linear Mappings with Values in a Dual......Page 329
15.4 Projective Limits and Projective Tensor Products......Page 331
15.5 Inductive Limits and Projective Tensor Products......Page 333
15.6 Some Stability Properties......Page 335
15.7 Projective Tensor Products with L_1(\mu)-spaces......Page 338
15.8 References......Page 341
16.1 \epsilon-Products and \epsilon-Tensor Products......Page 343
16.2 Tensor Product and Linear Mappings......Page 347
16.3 Projective and Inductive Limits......Page 350
16.4 Some Stability Properties......Page 353
16.5 Spaces of Summable Sequences......Page 357
16.6 Continuous Vector Valued Functions......Page 360
16.7 Holomorphic Vector Valued Functions......Page 362
16.8 References......Page 366
17.1 Compact Operators......Page 368
17.2 Weakly Compact Operators......Page 372
17.3 Nuclear Operators......Page 376
17.4 Integral Operators......Page 380
17.5 The Trace for Finite Operators......Page 386
17.6 Some Particular Cases......Page 391
17.7 References......Page 395
18.1 Generalities......Page 397
18.2 Some Stability Properties......Page 401
18.3 The Approximation Property for Banach Spaces......Page 403
18.4 The Metric Approximation Property......Page 408
18.5 The Approximation Property for Concrete Spaces......Page 410
18.6 References......Page 417
19.1 Generalities......Page 418
19.2 Dual, Injective, and Surjective Ideals......Page 420
19.3 Ideal-Quasinorms......Page 422
19.4 l_p-Sequences......Page 425
19.5 Absolutely p-Summing Operators......Page 428
19.6 Factorization......Page 431
19.7 p-Nuclear Operators......Page 434
19.8 p-Approximable Operators......Page 439
19.9 Strongly Nuclear Operators......Page 443
19.10 Some Multiplication Theorems......Page 445
19.11 References......Page 449
20.1 Compact Operators on Hilbert Spaces......Page 451
20.2 The Schatten-von Neumann Classes......Page 453
20.3 Grothendieck's Inequality......Page 458
20.4 Applications......Page 462
20.5 P_p and N_q on Hilbert Spaces......Page 467
20.6 Composition of Absolutely Summing Operators......Page 470
20.7 Weakly Compact Operators on C(K)-Spaces......Page 472
20.8 References......Page 476
21.1 Locally Convex A-Spaces......Page 478
21.2 Generalities on Nuclear Spaces......Page 482
21.3 Further Characterizations by Tensor Products......Page 486
21.4 Nuclear Spaces and Choquet Simplexes......Page 489
21.5 On Co-Nuclear Spaces......Page 491
21.6 Examples of Nuclear Spaces......Page 496
21.7 A Universal Generator......Page 500
21.8 Strongly Nuclear Spaces......Page 504
21.9 Associated Topologies......Page 508
21.10 Bases in Nuclear Spaces......Page 510
21.11 References......Page 517
Bibliography......Page 520
List of Symbols......Page 541
Index......Page 543