ISBN: 3540295860
TITLE: Infinite Dimensional Analysis
AUTHOR: Aliprantis/Border
TOC: 

Preface to the third edition vii
A foreword to the practical xix
1 Odds and ends 1
1.1 Numbers 1
1.2 Sets 2
1.3 Relations, correspondences, and functions 4
1.4 A bestiary of relations 5
1.5 Equivalence relations 7
1.6 Orders and such 7
1.7 Real functions 8
1.8 Duality of evaluation 9
1.9 Infinites 10
1.10 The Diagonal Theorem and Russell's Paradox 12
1.11 The axiom of choice and axiomatic set theory 13
1.12 Zorn's Lemma 15
1.13 Ordinals 18
2 Topology 21
2.1 Topological spaces 23
2.2 Neighbourhoods and closures 26
2.3 Dense subsets 28
2.4 Nets 29
2.5 Filters 32
2.6 Nets and Filters 35
2.7 Continuous functions 36
2.8 Compactness 38
2.9 Nets vs. sequences 41
2.10 Semicontinuous functions 43
2.11 Separation properties 44
2.12 Comparing topologies 47
2.13 Weak topologies 47
2.14 The product topology 50
2.15 Pointwise and uniform convergence 53
2.16 Locally compact spaces 55
2.17 The Stone-Cech compactification 58
2.18 Stone-Cech compactification of a discrete set 63
2.19 Paracompact spaces and partitions of unity 65
3 Metrizable spaces 69
3.1 Metricspaces 70
3.2 Completeness 73
3.3 Uniformly continuous functions 76
3.4 Semicontinuous functions on metric spaces 79
3.5 Distance functions 80
3.6 Embeddings and completions 84
3.7 Compactness and completeness 85
3.8 Countable products of metric spaces 89
3.9 The Hilbert cube and metrization 90
3.10 Locally compact metrizable spaces 92
3.11 The Baire Category Theorem 93
3.12 Contraction mappings 95
3.13 The Cantor set 98
3.14 The Baire space N^N 101
3.15 Uniformities 108
3.16 The Hausdorff distance 109
3.17 The Hausdorff metric topology 113
3.18 Topologies for spaces of subsets 119
3.19 The space C(X, Y) 123
4 Measurability 127
4.1 Algebras of sets 129
4.2 Rings and semirings of sets 131
4.3 Dynkin's lemma 135
4.4 The Borel \sigma-algebra 137
4.5 Measurable functions 139
4.6 The space of measurable functions 141
4.7 Simple functions 144
4.8 The \sigma-algebra induced by a function 147
4.9 Product structures 148
4.10 Carathodory functions 153
4.11 Borel functions and continuity 156
4.12 The Baire \sigma-algebra 158
5 Topological vector spaces 163
5.1 Linear topologies 166
5.2 Absorbing and circled sets 168
5.3 Metrizable topological vector spaces 172
5.4 The Open Mapping and Closed Graph Theorems 175
5.5 Finite dimensional topological vector spaces 177
5.6 Convexsets 181
5.7 Convex and concave functions 186
5.8 Sublinear functions and gauges 190
5.9 The Hahn-Banach Extension Theorem 195
5.10 Separating hyperplane theorems 197
5.11 Separation by continuous functionals 201
5.12 Locally convex spaces and seminorms 204
5.13 Separation in locally convex spaces 207
5.14 Dual pairs 211
5.15 Topologies consistent with a given dual 213
5.16 Polars 215
5.17 S-topologies 220
5.18 The Mackey topology 223
5.19 The strong topology 223
6 Normed spaces 225
6.1 Normed and Banach spaces 227
6.2 Linear operators on normed spaces 229
6.3 The norm dual of a normed space 230
6.4 The uniform boundedness principle 232
6.5 Weak topologies on normed spaces 235
6.6 Metrizability of weak topologies 237
6.7 Continuity of the evaluation 241
6.8 Adjoint operators 243
6.9 Projections and the fixed space of an operator 244
6.10 Hilbert spaces 246
7 Convexity 251
7.1 Extended-valued convex functions 254
7.2 Lower semicontinuous convex functions 255
7.3 Support points 258
7.4 Subgradients .264
7.5 Supporting hyperplanes and cones 268
7.6 Convex functions on finite dimensional spaces 271
7.7 Separation and support in finite dimensional spaces 275
7.8 Supporting convex subsets of Hilbert spaces 280
7.9 The Bishop-Phelps Theorem 281
7.10 Support functionals 288
7.11 Support functionals and the Hausdorff metric 292
7.12 Extreme points of convex sets 294
7.13 Quasiconvexity 299
7.14 Polytopes and weak neighbourhoods 300
7.15 Exposed points of convex sets 305
8 Riesz spaces 311
8.1 Orders, lattices, and cones 312
8.2 Riesz spaces 313
8.3 Order bounded sets 315
8.4 Order and lattice properties 316
8.5 The Riesz decomposition property 319
8.6 Disjointness 320
8.7 Riesz subspaces and ideals 321
8.8 Order convergence and order continuity 322
8.9 Bands 324
8.10 Positive functionals 325
8.11 Extending positive functionals 330
8.12 Positive operators 332
8.13 Topological Riesz spaces 334
8.14 The band generated by E' 339
8.15 Riesz pairs 340
8.16 Symmetric Riesz pairs 342
9 Banach lattices 347
9.1 Frchet and Banach lattices 348
9.2 The Stone-Weierstrass Theorem 352
9.3 Lattice homomorphisms and isometries 353
9.4 Order continuous norms 355
9.5 AM- and AL-spaces 357
9.6 The interior of the positive cone 362
9.7 Positive projections 364
9.8 The curious AL-space BV_0 365
10 Charges and measures 371
10.1 Set functions 374
10.2 Limits of sequences of measures 379
10.3 Outer measures and measurable sets 379
10.4 The CaratModory extension of a measure 381
10.5 Measure spaces 387
10.6 Lebesgue measure 389
10.7 Product measures 391
10.8 Measures an R" 392
10.9 Atoms .395
10.10 The AL-space of charges 396
10.11 The AL-space of measures 399
10.12 Absolute continuity 401
11 Integrals 403
11.1 The integral of a step function 404
11.2 Finitely additive integration of bounded functions 406
11.3 The Lebesgue integral 408
11.4 Continuity properties of the Lebesgue integral 413
11.5 The extended Lebesgue integral 416
11.6 Iterated integrals 418
11.7 The Riemann integral 419
11.8 The Bochner integral 422
11.9 The Gelfand integral 428
11.10 The Dunford and Pettis integrals 431
12 Measures and topology 433
12.1 Borel measures and regularity 434
12.2 Regular Borel measures 438
12.3 The support of a measure 441
12.4 Nonatomic Borel measures 443
12.5 Analytic sets 446
12.6 The Choquet Capacity Theorem 456
13 L_p-spaces 461
13.1 L_p-norms 462
13.2 Inequalities of Hlder and Minkowski 463
13.3 Dense subspaces of L_p-spaces 466
13.4 Sublattices of L_p-spaces 467
13.5 Separable L,-spaces and measures 468
13.6 The Radon-Nikodym Theorem 469
13.7 Equivalent measures 471
13.8 Duals of L_p-spaces 473
13.9 Lyapunov's Convexity Theorem 475
13.10 Convergence in measure 479
13.11 Convergence in measure in L_p-spaces 481
13.12 Change of variables 483
14 Riesz Representation Theorems 487
14.1 The AM-space B_b(\Sigma) and its dual 488
14.2 The dual of Cb(X) for normal spaces 491
14.3 The dual of Q(X) for locally compact spaces 496
14.4 Baire vs. Borel measures 498
14.5 Homomorphisms between C(X)-spaces 500
15 Probability measures 505
15.1 The weak* topology an P (X) 506
15.2 Embedding X in P (X) 512
15.3 Properties of P (X) 513
15.4 The many faces of P (X) 517
15.5 Compactness in P (X) 518
15.6 The Kolmogorov Extension Theorem 519
16 Spaces of sequences 525
16.1 The basic sequence spaces 526
16.2 The sequence spaces R^N and \phi 527
16.3 The sequence space c_0 529
16.4 The sequence space c 531
16.5 The l_p-spaces 533
16.6 l_1 and the symmetric Riesz pair <l_\infty,l_1> 537
16.7 The sequence space l_\infty 538
16.8 More an l_\infty = ba(N) 543
16.9 Embedding sequence spaces 546
16.10 Banach-Mazur limits and invariant measures 550
16.11 Sequences of vector spaces 552
17 Correspondences 555
17.1 Basic definitions 556
17.2 Continuity of correspondences 558
17.3 Hemicontinuity and nets 563
17.4 Operations on correspondences 566
17.5 The Maximum Theorem 569
17.6 Vector-valued correspondences 571
17.7 Demicontinuous correspondences 574
17.8 Knaster-Kuratowski-Mazurkiewicz mappings 577
17.9 Fixed point theorems 581
17.10 Contraction correspondences 585
17.11 Continuous selectors 587
18 Measurable correspondences 591
18.1 Measurability notions 592
18.2 Compact-valued correspondences as functions 597
18.3 Measurable selectors 600
18.4 Correspondences with measurable graph 606
18.5 Correspondences with compact convex values 609
18.6 Integration of correspondences 614
19 Markov transitions 621
19.1 Markov and stochastic operators 623
19.2 Markov transitions and kernels 625
19.3 Continuous Markov transitions 631
19.4 Invariant measures 631
19.5 Ergodic measures 636
19.6 Markov transition correspondences 638
19.7 Random functions 641
19.8 Dilations 645
19.9 More an Markov operators 650
19.10 A note an dynamical systems 652
20 Ergodicity 655
20.1 Measure-preserving transformations and ergodicity 656
20.2 Birkhoff's Ergodic Theorem 659
20.3 Ergodic operators 661
References 667
Index 681
END
