Topological Vector Spaces I

Front Cover......Page 1
Title Page......Page 4
Copyright Information......Page 5
Preface to the First Edition......Page 6
Preface to the Second Edition......Page 8
Contents......Page 10
1. The notion of a topological space......Page 18
2. Neighbourhoods......Page 19
4. Hausdorff spaces......Page 20
6. Induced topologies and comparison of topologies. Connectedness......Page 21
7. Continuous mappings......Page 23
8. Topological products......Page 24
2. Zorn's lemma......Page 26
3. Nets in topological spaces......Page 27
4. Filters......Page 28
5. Filters in topological spaces......Page 29
6. Nets and filters in topological products......Page 30
7. Ultrafilters......Page 31
8. Regular spaces......Page 32
1. Definition of compact spaces and sets......Page 33
2. Properties of compact sets......Page 34
4. Other concepts of compactness......Page 35
5. Axioms of countability......Page 36
6. Locally compact spaces......Page 37
7. Normal spaces......Page 39
2. Metric space as a topological space......Page 40
3. Continuity in metric spaces......Page 41
4. Completion of a metric space......Page 42
5. Separable and compact metric spaces......Page 43
6. Baire's theorem......Page 44
7. The topological product of metric spaces......Page 45
1. Definition......Page 46
2. The topology of a uniform space......Page 47
3. Uniform continuity......Page 48
4. Cauchy filters......Page 49
5. The completion of a Hausdorff uniform space......Page 50
6. Compact uniform spaces......Page 52
7. The product of uniform spaces......Page 54
1. Upper and lower limits......Page 55
2. Semi-continuous functions......Page 57
3. The least upper bound of a collection of functions......Page 58
4. Continuous functions on normal spaces......Page 59
6. Completely regular spaces......Page 61
7. Metrizable uniform spaces......Page 62
8. The complete regularity of uniform spaces......Page 64
1. Definition of a vector space......Page 65
3. Bases and complements......Page 67
4. The dimension of a linear space......Page 69
5. Isomorphism, canonical form......Page 70
6. Sums and intersections of subspaces......Page 71
7. Dimension and co-dimension of subspaces......Page 72
8. Products and direct sums of vector spaces......Page 73
9. Lattices......Page 74
10. The lattice of linear subspaces......Page 75
1. Definition and rules of calculation......Page 76
3. Projections......Page 77
4. Inverse mappings......Page 78
5. Representation by matrices......Page 80
6. Rings of matrices......Page 82
8. Canonical representation of a linear mapping......Page 83
9. Equivalence of mappings and matrices......Page 84
10. The theory of equivalence......Page 85
1. The dual space......Page 86
2. Orthogonality......Page 87
3. The lattice of orthogonally closed subspaces of E*......Page 89
4. The adjoint mapping......Page 90
5. The dimension of E*......Page 91
6. The tensor product of vector spaces......Page 93
7. Linear mappings of tensor products......Page 95
2. Linearly topologized spaces......Page 99
3. Dual pairs, weak topologies......Page 102
4. The dual space......Page 103
5. The dual pairs ......Page 105
6. Weak convergence and weak completeness......Page 106
7. Quotient spaces and topological complements......Page 107
8. Dual spaces of subspaces and quotient spaces......Page 110
9. Linearly compact spaces......Page 112
11. The topology T_lk......Page 114
12. T_lk-continuous linear mappings......Page 115
13. Continuous basis and continuous dimension......Page 117
1. The duality of E and E*......Page 118
2. The theory of the solutions of column- and row-finite systems of equations......Page 120
3. Formulae for solutions......Page 121
4. The countable case......Page 123
5. An example......Page 124
1. The structure of locally linearly compact spaces......Page 125
2. The endomorphisms of ψ......Page 126
3. The theory of equivalence in ψ......Page 128
1. Linearly bounded subspaces......Page 130
2. The linear strong topology......Page 131
3. The completion......Page 132
4. Topological sums and products......Page 134
5. Spaces of countable degree......Page 136
6. A counterexample......Page 137
7. Further investigations......Page 138
1. Definition of a normed space......Page 140
2. Norm isomorphism, equivalent norms......Page 142
3. Banach spaces......Page 143
4. Quotient spaces and topological products......Page 144
5. The dual space......Page 145
6. Continuous linear mappings......Page 146
7. The spaces c_0, c, l^1 and l^∞......Page 147
8. The spaces l^p, 19. (B)-spaces of continuous and holomorphic functions......Page 154
10. The Lp spaces (p > 1)......Page 156
11. The space L^∞......Page 159
1. Definition of a topological vector space......Page 161
2. A second definition......Page 163
3. The completion......Page 165
4. Quotient spaces and topological products......Page 166
5. Finite dimensional topological vector spaces......Page 168
6. Bounded and compact subsets......Page 169
8. Topologically complementary spaces......Page 172
9. The dual space, hyperplanes, the spaces Lp with 010. Locally bounded spaces, quasi-norms, p-norms......Page 176
11. Metrizable spaces......Page 179
12. The Banach-Schauder theorem and the closed-graph theorem......Page 183
13. Equicontinuous mappings, and the theorems of Banach and Banach-Steinhaus......Page 185
14. Bilinear mappings......Page 188
1. The convex and absolutely convex cover of a set......Page 190
2. The algebraic boundary of a convex set......Page 193
3. Half-spaces......Page 196
4. Convex bodies and the Minkowski functionals associated with them......Page 197
5. Convex cones......Page 200
6. Hypercones......Page 201
1. The separation theorem......Page 203
2. The Hahn-Banach theorem......Page 205
3. The analytic proof of the Hahn-Banach theorem......Page 206
4. Two consequences of the Hahn-Banach theorem......Page 209
5. Supporting hyperplanes......Page 210
6. The Hahn-Banach theorem for normed spaces. Adjoint mappings......Page 213
7. The dual space of C(I)......Page 214
1. Definition by neighbourhoods, and by semi-norms......Page 219
2. Metrizable locally convex spaces and (F)-spaces......Page 221
3. Subspaces, quotient spaces and topological products of locally convex spaces......Page 223
4. The completion of a locally convex space......Page 225
5. The locally convex direct sum of locally convex spaces......Page 228
1. The locally convex hull of locally convex spaces......Page 232
2. The inductive limit of vector spaces......Page 234
3. The topological inductive limit of locally convex spaces......Page 237
4. Strict inductive limits......Page 239
5. (LB)- and (LF)-spaces. Completeness......Page 240
6. The locally convex kernel of locally convex spaces......Page 242
7. The projective limit of vector spaces......Page 245
8. The topological projective limit of locally convex spaces......Page 247
9. The representation of a locally convex space as a projective limit......Page 248
10.  A criterion for completeness......Page 249
1. The existence of continuous linear functionals......Page 250
2. Dual pairs and weak topologies......Page 251
3. The duality of closed subspaces......Page 253
4. Duality of mappings......Page 254
5. Duality of complementary spaces......Page 255
6. The convex cover of a compact set......Page 257
7. The separation theorem for compact convex sets......Page 260
8. Polarity......Page 262
9. The polar of a neighbourhood of 0......Page 264
10. A representation of locally convex spaces......Page 266
11. Bounded and strongly bounded sets in dual pairs......Page 268
1. The topology T_M of uniform convergence on M......Page 271
2. The strong topology......Page 273
3. The original topology of a locally convex space; separability......Page 275
4. The Mackey topology......Page 277
5. The topology of a metrizable space......Page 279
6. The topology T_c of precompact convergence......Page 280
7. Polar topologies......Page 283
8. The topologies T^f and T^lf......Page 284
9. Grothendieck's construction of the completion......Page 286
10. The Banach-Dieudonne theorem......Page 289
11. Real and complex locally convex spaces......Page 290
1. The dual of subspaces and quotient spaces......Page 292
2. The topologies of subspaces, quotient spaces and their duals......Page 293
3. Subspaces and quotient spaces of normed spaces......Page 296
4. The quotient spaces of l^1......Page 297
5. The duality of topological products and locally convex direct sums......Page 300
6. The duality of locally convex hulls and kernels......Page 305
7. Topologies on locally convex hulls and kernels......Page 308
1. Quasi-completeness......Page 312
2. The bidual space......Page 314
3. Semi-reflexivity......Page 315
4. The topologies on the bidual......Page 317
5. Reflexivity......Page 319
6. The relationship between semi-reflexivity and reflexivity......Page 321
7. Distinguished spaces......Page 323
8. The dual of a semi-reflexive space......Page 324
9. Polar reflexivity......Page 325
1. The theorems of Smulian and Kaplansky......Page 327
2. Eberlein's theorem......Page 330
3. Further criteria for weak compactness......Page 332
4. Convex sets in spaces which are not semi-reflexive. The theorems of Klee......Page 336
5. Krein's theorem......Page 340
6. Ptak's theorem......Page 343
1. The Krein-Milman theorem......Page 347
2. Examples and applications......Page 350
3. Variants of the Krein-Milman theorem......Page 353
4. The extreme rays of a cone......Page 354
5. Locally compact convex sets......Page 356
1. Strict convexity......Page 359
2. Shortest distance......Page 360
3. Points of smoothness......Page 362
4. Weak differentiability of the norm......Page 364
5. Examples......Page 367
6. Uniform convexity......Page 370
7. The uniform convexity of the lp and Lp spaces......Page 372
8. Further examples......Page 376
9. Invariance under topological isomorphisms......Page 377
10. Uniform smoothness and strong differentiability of the norm......Page 380
11. Further ideas......Page 383
1. Quasi-barrelled spaces and barrelled spaces......Page 384
2. (M)-spaces and (FM)-spaces......Page 386
3. The space H(G)......Page 389
4. (M)-spaces of locally holomorphic functions......Page 392
1. Definition......Page 396
2. The structure of bornological spaces......Page 397
3. Local convergence. Sequentially continuous mappings......Page 399
4. Hereditary properties......Page 400
5. The dual, and the topology T_c_0......Page 401
6. Boundedly closed spaces......Page 403
7. Reflexivity and completeness......Page 405
8. The Mackey-Ulam theorem......Page 406
1. Fundamental sequences of bounded sets. Metrizability......Page 409
2. The bidual......Page 411
3. (DF)-spaces......Page 413
4. Bornological (DF)-spaces......Page 416
5. Hereditary properties of (DF)-spaces......Page 418
6. Further results, and open questions......Page 420
1. The α-dual. Examples......Page 422
2. The normal topology of a sequence space......Page 424
3. Sums and products of sequence spaces......Page 426
4. Unions and intersections of sequence spaces......Page 427
5. Topological properties of sequence spaces......Page 429
6. Compact subsets of a perfect space......Page 432
7. Barrelled spaces and (M)-spaces......Page 434
8. Echelon and co-echelon spaces......Page 436
9. Co-echelon spaces of type (M)......Page 438
10. Further investigations into sequence spaces......Page 440
1. The dual of l^∞......Page 441
2. Subspaces of l^∞ and l^1 with no topological complements......Page 443
3. The problem of complements in l^p and Lp......Page 445
4. Complements in (F)-spaces......Page 448
5. An (FM)-space......Page 450
6. An (LB)-space which is not complete......Page 451
7. An (F)-space which is not distinguished......Page 452
Bibliography......Page 454
Author and Subject Index......Page 464