Aliprantis / Border Infinite Dimensional Analysis

A Hitchhiker's Guide
3rd Auflage 2006
ISBN: 978-3-540-29587-7
Verlag: Springer Berlin Heidelberg
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 704 Seiten

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.

Autoren/Hrsg.

Aliprantis, Charalambos D.

Border, Kim C.

Inhaltsverzeichnis

1;Prefaces;6
1.1;Preface to the third edition;6
1.2;Preface to the second edition;7
1.3;Preface to the first edition;8
2;Contents;10
3;A foreword to the practical;17
3.1;Why use infinite dimensional analysis?;17
3.2;Spaces of sequences;18
3.3;Spaces of functions;18
3.4;Spaces of measures;19
3.5;Spaces of sets;19
3.6;Prerequisites;20
4;1 Odds and ends;21
4.1;1.1 Numbers;21
4.2;1.2 Sets;22
4.3;1.3 Relations, correspondences, and functions;24
4.4;1.4 A bestiary of relations;25
4.5;1.5 Equivalence relations;27
4.6;1.6 Orders and such;27
4.7;1.7 Real functions;28
4.8;1.8 Duality of evaluation;29
4.9;1.9 In. nit\/ yies;30
4.10;1.10 The Diagonal Theorem and Russell’s Paradox;32
4.11;1.11 The axiom of choice and axiomatic set theory;33
4.12;1.12 Zorn’s Lemma;35
4.13;1.13 Ordinals;38
5;2 Topology;41
5.1;2.1 Topological spaces;43
5.2;2.2 Neighborhoods and closures;46
5.3;2.3 Dense subsets;48
5.4;2.4 Nets;49
5.5;2.5 Filters;52
5.6;2.6 Nets and Filters;55
5.7;2.7 Continuous functions;56
5.8;2.8 Compactness;58
5.9;2.9 Nets vs. sequences;61
5.10;2.10 Semicontinuous functions;63
5.11;2.11 Separation properties;64
5.12;2.12 Comparing topologies;67
5.13;2.13 Weak topologies;67
5.14;2.14 The product topology;70
5.15;2.15 Pointwise and uniform convergence;73
5.16;2.16 Locally compact spaces;75
5.17;2.17 The Stone–Cech compacti.cation;78
5.18;2.18 Stone–Cech compacti.cation of a discrete set;83
5.19;2.19 Paracompact spaces and partitions of unity;85
6;3 Metrizable spaces;88
6.1;3.1 Metric spaces;89
6.2;3.2 Completeness;92
6.3;3.3 Uniformly continuous functions;95
6.4;3.4 Semicontinuous functions on metric spaces;98
6.5;3.5 Distance functions;99
6.6;3.6 Embeddings and completions;103
6.7;3.7 Compactness and completeness;104
6.8;3.8 Countable products of metric spaces;108
6.9;3.9 The Hilbert cube and metrization;109
6.10;3.10 Locally compact metrizable spaces;111
6.11;3.11 The Baire Category Theorem;112
6.12;3.12 Contraction mappings;114
6.13;3.13 The Cantor set;117
6.14;3.14 The Baire space;120
6.15;3.15 Uniformities;127
6.16;3.16 The Hausdorff Distance;128
6.17;3.17 The Hausdorff metric topology;132
6.18;3.18 Topologies for spaces of subsets;138
6.19;3.19 The space;142
7;4 Measurability;146
7.1;4.1 Algebras of sets;148
7.2;4.2 Rings and semirings of sets;150
7.3;4.3 Dynkin’s lemma;154
7.4;4.4 The Borel s-algebra;156
7.5;4.5 Measurable functions;158
7.6;4.6 The space of measurable functions;160
7.7;4.7 Simple functions;163
7.8;4.8 The s-algebra induced by a function;166
7.9;4.9 Product structures;167
7.10;4.10 Carathéodory functions;172
7.11;4.11 Borel functions and continuity;175
7.12;4.12 The Baire s-algebra;177
8;5 Topological vector spaces;181
8.1;5.1 Linear topologies;184
8.2;5.2 Absorbing and circled sets;186
8.3;5.3 Metrizable topological vector spaces;190
8.4;5.4 The Open Mapping and Closed Graph Theorems;193
8.5;5.5 Finite dimensional topological vector spaces;195
8.6;5.6 Convex sets;199
8.7;5.7 Convex and concave functions;204
8.8;5.8 Sublinear functions and gauges;208
8.9;5.9 The Hahn–Banach Extension Theorem;213
8.10;5.10 Separating hyperplane theorems;215
8.11;5.11 Separation by continuous functionals;219
8.12;5.12 Locally convex spaces and seminorms;222
8.13;5.13 Separation in locally convex spaces;225
8.14;5.14 Dual pairs;229
8.15;5.15 Topologies consistent with a given dual;231
8.16;5.16 Polars;233
8.17;5.17 S-topologies;238
8.18;5.18 The Mackey topology;241
8.19;5.19 The strong topology;241
9;6 Normed spaces;243
9.1;6.1 Normed and Banach spaces;245
9.2;6.2 Linear operators on normed spaces;247
9.3;6.3 The norm dual of a normed space;248
9.4;6.4 The uniform boundedness principle;250
9.5;6.5 Weak topologies on normed spaces;253
9.6;6.6 Metrizability of weak topologies;255
9.7;6.7 Continuity of the evaluation;259
9.8;6.8 Adjoint operators;261
9.9;6.9 Projections and the .xed space of an operator;262
9.10;6.10 Hilbert spaces;264
10;7 Convexity;269
10.1;7.1 Extended-valued convex functions;272
10.2;7.2 Lower semicontinuous convex functions;273
10.3;7.3 Support points;276
10.4;7.4 Subgradients;282
10.5;7.5 Supporting hyperplanes and cones;286
10.6;7.6 Convex functions on .nite dimensional spaces;289
10.7;7.7 Separation and support in .nite dimensional spaces;293
10.8;7.8 Supporting convex subsets of Hilbert spaces;298
10.9;7.9 The Bishop–Phelps Theorem;299
10.10;7.10 Support functionals;306
10.11;7.11 Support functionals and the Hausdorff metric;310
10.12;7.12 Extreme points of convex sets;312
10.13;7.13 Quasiconvexity;317
10.14;7.14 Polytopes and weak neighborhoods;318
10.15;7.15 Exposed points of convex sets;323
11;8 Riesz spaces;328
11.1;8.1 Orders, lattices, and cones;329
11.2;8.2 Riesz spaces;330
11.3;8.3 Order bounded sets;332
11.4;8.4 Order and lattice properties;333
11.5;8.5 The Riesz decomposition property;336
11.6;8.6 Disjointness;337
11.7;8.7 Riesz subspaces and ideals;338
11.8;8.8 Order convergence and order continuity;339
11.9;8.9 Bands;341
11.10;8.10 Positive functionals;342
11.11;8.11 Extending positive functionals;347
11.12;8.12 Positive operators;349
11.13;8.13 Topological Riesz spaces;351
11.14;8.14 The band generated by;356
11.15;8.15 Riesz pairs;357
11.16;8.16 Symmetric Riesz pairs;359
12;9 Banach lattices;363
12.1;9.1 Fréchet and Banach lattices;364
12.2;9.2 The Stone–Weierstrass Theorem;368
12.3;9.3 Lattice homomorphisms and isometries;369
12.4;9.4 Order continuous norms;371
12.5;9.5 AM- and AL-spaces;373
12.6;9.6 The interior of the positive cone;378
12.7;9.7 Positive projections;380
12.8;9.8 The curious AL-space;381
13;10 Charges and measures;387
13.1;10.1 Set functions;390
13.2;10.2 Limits of sequences of measures;395
13.3;10.3 Outer measures and measurable sets;395
13.4;10.4 The Carathéodory extension of a measure;397
13.5;10.5 Measure spaces;403
13.6;10.6 Lebesgue measure;405
13.7;10.7 Product measures;407
13.8;10.8 Measures on;408
13.9;10.9 Atoms;411
13.10;10.10 The AL-space of charges;412
13.11;10.11 The AL-space of measures;415
13.12;10.12 Absolute continuity;417
14;11 Integrals;419
14.1;11.1 The integral of a step function;420
14.2;11.2 Finitely additive integration of bounded functions;422
14.3;11.3 The Lebesgue integral;424
14.4;11.4 Continuity properties of the Lebesgue integral;429
14.5;11.5 The extended Lebesgue integral;432
14.6;11.6 Iterated integrals;434
14.7;11.7 The Riemann integral;435
14.8;11.8 The Bochner integral;438
14.9;11.9 The Gelfand integral;444
14.10;11.10 The Dunford and Pettis integrals;447
15;12 Measures and topology;449
15.1;12.1 Borel measures and regularity;450
15.2;12.2 Regular Borel measures;454
15.3;12.3 The support of a measure;457
15.4;12.4 Nonatomic Borel measures;459
15.5;12.5 Analytic sets;462
15.6;12.6 The Choquet Capacity Theorem;472
16;13 Lp-spaces;477
16.1;13.1 Lp-norms;478
16.2;13.2 Inequalities of Hölder and Minkowski;479
16.3;13.3 Dense subspaces of Lp-spaces;482
16.4;13.4 Sublattices of Lp-spaces;483
16.5;13.5 Separable;484
16.6;spaces and measures;484
16.7;13.6 The Radon–Nikodym Theorem;485
16.8;13.7 Equivalent measures;487
16.9;13.8 Duals of Lp-spaces;489
16.10;13.9 Lyapunov’s Convexity Theorem;491
16.11;13.10 Convergence in measure;495
16.12;13.11 Convergence in measure in lp-spaces;497
16.13;13.12 Change of variables;499
17;14 Riesz Representation Theorems;503
17.1;14.1 The AM-space Bb(S) and its dual;504
17.2;14.2 The dual of Cb(X) for normal spaces;507
17.3;14.3 The dual of Cc(X) for locally compact spaces;512
17.4;14.4 Baire vs. Borel measures;514
17.5;14.5 Homomorphisms between C(X)-spaces;516
18;15 Probability measures;520
18.1;15.1 The weak* topology on;521
18.2;15.2 Embedding X in P (X);527
18.3;15.3 Properties of P (X);528
18.4;15.4 The many faces of P (X);532
18.5;15.5 Compactness in P (X);533
18.6;15.6 The Kolmogorov Extension Theorem;534
19;16 Spaces of sequences;539
19.1;16.1 The basic sequence spaces;540
19.2;16.2 The sequence spaces RN and .;541
19.3;16.3 The sequence space;543
19.4;16.4 The sequence space;545
19.5;16.5 The p-spaces;547
19.6;16.6 and the symmetric Riesz pair;551
19.7;16.7 The sequence space;552
19.8;16.8 More on;557
19.9;16.9 Embedding sequence spaces;560
19.10;16.10 Banach–Mazur limits and invariant measures;564
19.11;16.11 Sequences of vector spaces;566
20;17 Correspondences;569
20.1;17.1 Basic de.nitions;570
20.2;17.2 Continuity of correspondences;572
20.3;17.3 Hemicontinuity and nets;577
20.4;17.4 Operations on correspondences;580
20.5;17.5 The Maximum Theorem;583
20.6;17.6 Vector-valued correspondences;585
20.7;17.7 Demicontinuous correspondences;588
20.8;17.8 Knaster –Kuratowski–Mazurkiewicz mappings;591
20.9;17.9 Fixed point theorems;595
20.10;17.10 Contraction correspondences;599
20.11;17.11 Continuous selectors;601
21;18 Measurable correspondences;605
21.1;18.1 Measurability notions;606
21.2;18.2 Compact-valued correspondences as functions;611
21.3;18.3 Measurable selectors;614
21.4;18.4 Correspondences with measurable graph;620
21.5;18.5 Correspondences with compact convex values;623
21.6;18.6 Integration of correspondences;628
22;19 Markov transitions;635
22.1;19.1 Markov and stochastic operators;637
22.2;19.2 Markov transitions and kernels;639
22.3;19.3 Continuous Markov transitions;645
22.4;19.4 Invariant measures;645
22.5;19.5 Ergodic measures;650
22.6;19.6 Markov transition correspondences;652
22.7;19.7 Random functions;655
22.8;19.8 Dilations;659
22.9;19.9 More on Markov operators;664
22.10;19.10 A note on dynamical systems;666
23;20 Ergodicity;668
23.1;20.1 Measure-preserving transformations and ergodicity;669
23.2;20.2 Birkho;672
23.3;20.3 Ergodic operators;674
24;References;680
25;Index;694