Infinite Dimensional Analysis (eBook)

A Hitchhiker's Guide

Charalambos D. Aliprantis, Kim C. Border (Autoren)

eBook Download: PDF

2006 | 3rd ed. 2006
XXII, 704 Seiten
Springer Berlin (Verlag)
978-3-540-29587-7 (ISBN)

Infos

What you'll find in this monograph is nothing less than a complete and rigorous study of modern functional analysis. It is intended for the student or researcher who could benefit from functional analytic methods, but who does not have an extensive background in the subject and does not plan to make a career as a functional analyst. It develops the topological structures in connection with a number of topic areas such as measure theory, convexity, and Banach lattices, as well as covering the analytic approach to Markov processes. Many of the results were previously available only in works scattered throughout the literature.

Inhalt

Prefaces 6
Preface to the third edition 6
Preface to the second edition 7
Preface to the first edition 8
Contents 10
A foreword to the practical 17
Why use infinite dimensional analysis? 17
Spaces of sequences 18
Spaces of functions 18
Spaces of measures 19
Spaces of sets 19
Prerequisites 20
1 Odds and ends 21
1.1 Numbers 21
1.2 Sets 22
1.3 Relations, correspondences, and functions 24
1.4 A bestiary of relations 25
1.5 Equivalence relations 27
1.6 Orders and such 27
1.7 Real functions 28
1.8 Duality of evaluation 29
1.9 In. nit// yies 30
1.10 The Diagonal Theorem and Russell’s Paradox 32
1.11 The axiom of choice and axiomatic set theory 33
1.12 Zorn’s Lemma 35
1.13 Ordinals 38
2 Topology 41
2.1 Topological spaces 43
2.2 Neighborhoods and closures 46
2.3 Dense subsets 48
2.4 Nets 49
2.5 Filters 52
2.6 Nets and Filters 55
2.7 Continuous functions 56
2.8 Compactness 58
2.9 Nets vs. sequences 61
2.10 Semicontinuous functions 63
2.11 Separation properties 64
2.12 Comparing topologies 67
2.13 Weak topologies 67
2.14 The product topology 70
2.15 Pointwise and uniform convergence 73
2.16 Locally compact spaces 75
2.17 The Stone–Cech compacti.cation 78
2.18 Stone–Cech compacti.cation of a discrete set 83
2.19 Paracompact spaces and partitions of unity 85
3 Metrizable spaces 88
3.1 Metric spaces 89
3.2 Completeness 92
3.3 Uniformly continuous functions 95
3.4 Semicontinuous functions on metric spaces 98
3.5 Distance functions 99
3.6 Embeddings and completions 103
3.7 Compactness and completeness 104
3.8 Countable products of metric spaces 108
3.9 The Hilbert cube and metrization 109
3.10 Locally compact metrizable spaces 111
3.11 The Baire Category Theorem 112
3.12 Contraction mappings 114
3.13 The Cantor set 117
3.14 The Baire space 120
3.15 Uniformities 127
3.16 The Hausdorff Distance 128
3.17 The Hausdorff metric topology 132
3.18 Topologies for spaces of subsets 138
3.19 The space 142
4 Measurability 146
4.1 Algebras of sets 148
4.2 Rings and semirings of sets 150
4.3 Dynkin’s lemma 154
4.4 The Borel s-algebra 156
4.5 Measurable functions 158
4.6 The space of measurable functions 160
4.7 Simple functions 163
4.8 The s-algebra induced by a function 166
4.9 Product structures 167
4.10 Carathéodory functions 172
4.11 Borel functions and continuity 175
4.12 The Baire s-algebra 177
5 Topological vector spaces 181
5.1 Linear topologies 184
5.2 Absorbing and circled sets 186
5.3 Metrizable topological vector spaces 190
5.4 The Open Mapping and Closed Graph Theorems 193
5.5 Finite dimensional topological vector spaces 195
5.6 Convex sets 199
5.7 Convex and concave functions 204
5.8 Sublinear functions and gauges 208
5.9 The Hahn–Banach Extension Theorem 213
5.10 Separating hyperplane theorems 215
5.11 Separation by continuous functionals 219
5.12 Locally convex spaces and seminorms 222
5.13 Separation in locally convex spaces 225
5.14 Dual pairs 229
5.15 Topologies consistent with a given dual 231
5.16 Polars 233
5.17 S-topologies 238
5.18 The Mackey topology 241
5.19 The strong topology 241
6 Normed spaces 243
6.1 Normed and Banach spaces 245
6.2 Linear operators on normed spaces 247
6.3 The norm dual of a normed space 248
6.4 The uniform boundedness principle 250
6.5 Weak topologies on normed spaces 253
6.6 Metrizability of weak topologies 255
6.7 Continuity of the evaluation 259
6.8 Adjoint operators 261
6.9 Projections and the .xed space of an operator 262
6.10 Hilbert spaces 264
7 Convexity 269
7.1 Extended-valued convex functions 272
7.2 Lower semicontinuous convex functions 273
7.3 Support points 276
7.4 Subgradients 282
7.5 Supporting hyperplanes and cones 286
7.6 Convex functions on .nite dimensional spaces 289
7.7 Separation and support in .nite dimensional spaces 293
7.8 Supporting convex subsets of Hilbert spaces 298
7.9 The Bishop–Phelps Theorem 299
7.10 Support functionals 306
7.11 Support functionals and the Hausdorff metric 310
7.12 Extreme points of convex sets 312
7.13 Quasiconvexity 317
7.14 Polytopes and weak neighborhoods 318
7.15 Exposed points of convex sets 323
8 Riesz spaces 328
8.1 Orders, lattices, and cones 329
8.2 Riesz spaces 330
8.3 Order bounded sets 332
8.4 Order and lattice properties 333
8.5 The Riesz decomposition property 336
8.6 Disjointness 337
8.7 Riesz subspaces and ideals 338
8.8 Order convergence and order continuity 339
8.9 Bands 341
8.10 Positive functionals 342
8.11 Extending positive functionals 347
8.12 Positive operators 349
8.13 Topological Riesz spaces 351
8.14 The band generated by 356
8.15 Riesz pairs 357
8.16 Symmetric Riesz pairs 359
9 Banach lattices 363
9.1 Fréchet and Banach lattices 364
9.2 The Stone–Weierstrass Theorem 368
9.3 Lattice homomorphisms and isometries 369
9.4 Order continuous norms 371
9.5 AM- and AL-spaces 373
9.6 The interior of the positive cone 378
9.7 Positive projections 380
9.8 The curious AL-space 381
10 Charges and measures 387
10.1 Set functions 390
10.2 Limits of sequences of measures 395
10.3 Outer measures and measurable sets 395
10.4 The Carathéodory extension of a measure 397
10.5 Measure spaces 403
10.6 Lebesgue measure 405
10.7 Product measures 407
10.8 Measures on 408
10.9 Atoms 411
10.10 The AL-space of charges 412
10.11 The AL-space of measures 415
10.12 Absolute continuity 417
11 Integrals 419
11.1 The integral of a step function 420
11.2 Finitely additive integration of bounded functions 422
11.3 The Lebesgue integral 424
11.4 Continuity properties of the Lebesgue integral 429
11.5 The extended Lebesgue integral 432
11.6 Iterated integrals 434
11.7 The Riemann integral 435
11.8 The Bochner integral 438
11.9 The Gelfand integral 444
11.10 The Dunford and Pettis integrals 447
12 Measures and topology 449
12.1 Borel measures and regularity 450
12.2 Regular Borel measures 454
12.3 The support of a measure 457
12.4 Nonatomic Borel measures 459
12.5 Analytic sets 462
12.6 The Choquet Capacity Theorem 472
13 Lp-spaces 477
13.1 Lp-norms 478
13.2 Inequalities of Hölder and Minkowski 479
13.3 Dense subspaces of Lp-spaces 482
13.4 Sublattices of Lp-spaces 483
13.5 Separable 484
spaces and measures 484
13.6 The Radon–Nikodym Theorem 485
13.7 Equivalent measures 487
13.8 Duals of Lp-spaces 489
13.9 Lyapunov’s Convexity Theorem 491
13.10 Convergence in measure 495
13.11 Convergence in measure in lp-spaces 497
13.12 Change of variables 499
14 Riesz Representation Theorems 503
14.1 The AM-space Bb(S) and its dual 504
14.2 The dual of Cb(X) for normal spaces 507
14.3 The dual of Cc(X) for locally compact spaces 512
14.4 Baire vs. Borel measures 514
14.5 Homomorphisms between C(X)-spaces 516
15 Probability measures 520
15.1 The weak* topology on 521
15.2 Embedding X in P (X) 527
15.3 Properties of P (X) 528
15.4 The many faces of P (X) 532
15.5 Compactness in P (X) 533
15.6 The Kolmogorov Extension Theorem 534
16 Spaces of sequences 539
16.1 The basic sequence spaces 540
16.2 The sequence spaces RN and . 541
16.3 The sequence space 543
16.4 The sequence space 545
16.5 The p-spaces 547
16.6 and the symmetric Riesz pair 551
16.7 The sequence space 552
16.8 More on 557
16.9 Embedding sequence spaces 560
16.10 Banach–Mazur limits and invariant measures 564
16.11 Sequences of vector spaces 566
17 Correspondences 569
17.1 Basic de.nitions 570
17.2 Continuity of correspondences 572
17.3 Hemicontinuity and nets 577
17.4 Operations on correspondences 580
17.5 The Maximum Theorem 583
17.6 Vector-valued correspondences 585
17.7 Demicontinuous correspondences 588
17.8 Knaster –Kuratowski–Mazurkiewicz mappings 591
17.9 Fixed point theorems 595
17.10 Contraction correspondences 599
17.11 Continuous selectors 601
18 Measurable correspondences 605
18.1 Measurability notions 606
18.2 Compact-valued correspondences as functions 611
18.3 Measurable selectors 614
18.4 Correspondences with measurable graph 620
18.5 Correspondences with compact convex values 623
18.6 Integration of correspondences 628
19 Markov transitions 635
19.1 Markov and stochastic operators 637
19.2 Markov transitions and kernels 639
19.3 Continuous Markov transitions 645
19.4 Invariant measures 645
19.5 Ergodic measures 650
19.6 Markov transition correspondences 652
19.7 Random functions 655
19.8 Dilations 659
19.9 More on Markov operators 664
19.10 A note on dynamical systems 666
20 Ergodicity 668
20.1 Measure-preserving transformations and ergodicity 669
20.2 Birkho 672
20.3 Ergodic operators 674
References 680
Index 694

Produkt-Details

• Erscheint lt. Verlag: 8.8.2006
• Zusatzinfo: XXII, 704 p.
• Verlagsort: Berlin
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik
• Themenwelt: Technik
• Themenwelt: Wirtschaft ► Volkswirtschaftslehre
• Schlagworte: Analysis • Convex Analysis • Economic Theory • Functional Analysis • linear optimization • Mathematical Analysis • Measure Theory and Integration • Theory of Finance • Topology
• ISBN-10: 3-540-29587-9 / 3540295879
• ISBN-13: 978-3-540-29587-7 / 9783540295877