Infinite Dimensional Analysis

A Hitchhiker's Guide

By: Charalambos D. Aliprantis, Kim C. Border

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Paperback | 2 May 2007 | Edition Number 3

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728 Pages

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This new edition of The Hitchhiker's Guide has bene?tted from the comments of many individuals, which have resulted in the addition of some new material, and the reorganization of some of the rest. The most obvious change is the creation of a separate Chapter 7 on convex analysis. Parts of this chapter appeared in elsewhere in the second edition, but much of it is new to the third edition. In particular, there is an expanded discussion of support points of convex sets, and a new section on subgradients of convex functions. There is much more material on the special properties of convex sets and functions in ?nite dimensional spaces. There are improvements and additions in almost every chapter. There is more new material than might seem at ?rst glance, thanks to a change in font that - duced the page count about ?ve percent. We owe a huge debt to Valentina Galvani, Daniela Puzzello, and Francesco Rusticci, who were participants in a graduate seminar at Purdue University and whose suggestions led to many improvements, especially in chapters ?ve through eight. We particularly thank Daniela Puzzello for catching uncountably many errors throughout the second edition, and simplifying the statements of several theorems and proofs. In another graduate seminar at Caltech, many improvements and corrections were suggested by Joel Grus, PJ Healy, Kevin Roust, Maggie Penn, and Bryan Rogers.

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Non-Fiction Mathematics Calculus & Mathematical Analysis Functional Analysis & Transforms Economics Economic Theory & Philosophy Applied Mathematics Optimisation Game Theory

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Published: 19th May 2006

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Preface to the third edition	p. vii
A foreword to the practical	p. xix
Odds and ends	p. 1
Numbers	p. 1
Sets	p. 2
Relations, correspondences, and functions	p. 4
A bestiary of relations	p. 5
Equivalence relations	p. 7
Orders and such	p. 7
Real functions	p. 8
Duality of evaluation	p. 9
Infinities	p. 10
The Diagonal Theorem and Russell's Paradox	p. 12
The axiom of choice and axiomatic set theory	p. 13
Zorn's Lemma	p. 15
Ordinals	p. 18
Topology	p. 21
Topological spaces	p. 23
Neighborhoods and closures	p. 26
Dense subsets	p. 28
Nets	p. 29
Filters	p. 32
Nets and Filters	p. 35
Continuous functions	p. 36
Compactness	p. 38
Nets vs. sequences	p. 41
Semicontinuous functions	p. 43
Separation properties	p. 44
Comparing topologies	p. 47
Weak topologies	p. 47
The product topology	p. 50
Pointwise and uniform convergence	p. 53
Locally compact spaces	p. 55
The Stone-Cech compactification	p. 58
Stone-Cech compactification of a discrete set	p. 63
Paracompact spaces and partitions of unity	p. 65
Metrizable spaces	p. 69
Metric spaces	p. 70
Completeness	p. 73
Uniformly continuous functions	p. 76
Semicontinuous functions on metric spaces	p. 79
Distance functions	p. 80
Embeddings and completions	p. 84
Compactness and completeness	p. 85
Countable products of metric spaces	p. 89
The Hilbert cube and metrization	p. 90
Locally compact metrizable spaces	p. 92
The Baire Category Theorem	p. 93
Contraction mappings	p. 95
The Cantor set	p. 98
The Baire space N[superscript N]	p. 101
Uniformities	p. 108
The Hausdorff distance	p. 109
The Hausdorff metric topology	p. 113
Topologies for spaces of subsets	p. 119
The space C(X, Y)	p. 123
Measurability	p. 127
Algebras of sets	p. 129
Rings and semirings of sets	p. 131
Dynkin's lemma	p. 135
The Borel [sigma]-algebra	p. 137
Measurable functions	p. 139
The space of measurable functions	p. 141
Simple functions	p. 144
The [sigma]-algebra induced by a function	p. 147
Product structures	p. 148
Caratheodory functions	p. 153
Borel functions and continuity	p. 156
The Baire [sigma]-algebra	p. 158
Topological vector spaces	p. 163
Linear topologies	p. 166
Absorbing and circled sets	p. 168
Metrizable topological vector spaces	p. 172
The Open Mapping and Closed Graph Theorems	p. 175
Finite dimensional topological vector spaces	p. 177
Convex sets	p. 181
Convex and concave functions	p. 186
Sublinear functions and gauges	p. 190
The Hahn-Banach Extension Theorem	p. 195
Separating hyperplane theorems	p. 197
Separation by continuous functionals	p. 201
Locally convex spaces and seminorms	p. 204
Separation in locally convex spaces	p. 207
Dual pairs	p. 211
Topologies consistent with a given dual	p. 213
Polars	p. 215
S-topologies	p. 220
The Mackey topology	p. 223
The strong topology	p. 223
Normed spaces	p. 225
Normed and Banach spaces	p. 227
Linear operators on normed spaces	p. 229
The norm dual of a normed space	p. 230
The uniform boundedness principle	p. 232
Weak topologies on normed spaces	p. 235
Metrizability of weak topologies	p. 237
Continuity of the evaluation	p. 241
Adjoint operators	p. 243
Projections and the fixed space of an operator	p. 244
Hilbert spaces	p. 246
Convexity	p. 251
Extended-valued convex functions	p. 254
Lower semicontinuous convex functions	p. 255
Support points	p. 258
Subgradients	p. 264
Supporting hyperplanes and cones	p. 268
Convex functions on finite dimensional spaces	p. 271
Separation and support in finite dimensional spaces	p. 275
Supporting convex subsets of Hilbert spaces	p. 280
The Bishop-Phelps Theorem	p. 281
Support functionals	p. 288
Support functionals and the Hausdorff metric	p. 292
Extreme points of convex sets	p. 294
Quasiconvexity	p. 299
Polytopes and weak neighborhoods	p. 300
Exposed points of convex sets	p. 305
Riesz spaces	p. 311
Orders, lattices, and cones	p. 312
Riesz spaces	p. 313
Order bounded sets	p. 315
Order and lattice properties	p. 316
The Riesz decomposition property	p. 319
Disjointness	p. 320
Riesz subspaces and ideals	p. 321
Order convergence and order continuity	p. 322
Bands	p. 324
Positive functionals	p. 325
Extending positive functionals	p. 330
Positive operators	p. 332
Topological Riesz spaces	p. 334
The band generated by E[prime]	p. 339
Riesz pairs	p. 340
Symmetric Riesz pairs	p. 342
Banach lattices	p. 347
Frechet and Banach lattices	p. 348
The Stone-Weierstrass Theorem	p. 352
Lattice homomorphisms and isometries	p. 353
Order continuous norms	p. 355
AM- and AL-spaces	p. 357
The interior of the positive cone	p. 362
Positive projections	p. 364
The curious AL-space BV[subscript 0]	p. 365
Charges and measures	p. 371
Set functions	p. 374
Limits of sequences of measures	p. 379
Outer measures and measurable sets	p. 379
The Caratheodory extension of a measure	p. 381
Measure spaces	p. 387
Lebesgue measure	p. 389
Product measures	p. 391
Measures on R[superscript n]	p. 392
Atoms	p. 395
The AL-space of charges	p. 396
The AL-space of measures	p. 399
Absolute continuity	p. 401
Integrals	p. 403
The integral of a step function	p. 404
Finitely additive integration of bounded functions	p. 406
The Lebesgue integral	p. 408
Continuity properties of the Lebesgue integral	p. 413
The extended Lebesgue integral	p. 416
Iterated integrals	p. 418
The Riemann integral	p. 419
The Bochner integral	p. 422
The Gelfand integral	p. 428
The Dunford and Pettis integrals	p. 431
Measures and topology	p. 433
Borel measures and regularity	p. 434
Regular Borel measures	p. 438
The support of a measure	p. 441
Nonatomic Borel measures	p. 443
Analytic sets	p. 446
The Choquet Capacity Theorem	p. 456
L[subscript p]-spaces	p. 461
L[subscript p]-norms	p. 462
Inequalities of Holder and Minkowski	p. 463
Dense subspaces of L[subscript p]-spaces	p. 466
Sublattices of L[subscript p]-spaces	p. 467
Separable L[subscript 1]-spaces and measures	p. 468
The Radon-Nikodym Theorem	p. 469
Equivalent measures	p. 471
Duals of L[subscript p]-spaces	p. 473
Lyapunov's Convexity Theorem	p. 475
Convergence in measure	p. 479
Convergence in measure in L[subscript p]-spaces	p. 481
Change of variables	p. 483
Riesz Representation Theorems	p. 487
The AM-space B[subscript b]([Sum]) and its dual	p. 488
The dual of C[subscript b](X) for normal spaces	p. 491
The dual of C[subscript c](X) for locally compact spaces	p. 496
Baire vs. Borel measures	p. 498
Homomorphisms between C(X)-spaces	p. 500
Probability measures	p. 505
The weak topology on P (X)	p. 506
Embedding X in P (X)	p. 512
Properties of P (X)	p. 513
The many faces of P (X)	p. 517
Compactness in P (X)	p. 518
The Kolmogorov Extension Theorem	p. 519
Spaces of sequences	p. 525
The basic sequence spaces	p. 526
The sequence spaces R[superscript N] and [phi]	p. 527
The sequence space c[subscript 0]	p. 529
The sequence space c	p. 531
The l[subscript p]-spaces	p. 533
l][subscript 1] and the symmetric Riesz pair [left angle bracket]l[subscript infinity], l[subscript 1][right angle bracket]	p. 537
The sequence space l[subscript infinity]	p. 538
More on l[Characters not reproducible] = ba(N)	p. 543
Embedding sequence spaces	p. 546
Banach-Mazur limits and invariant measures	p. 550
Sequences of vector spaces	p. 552
Correspondences	p. 555
Basic definitions	p. 556
Continuity of correspondences	p. 558
Hemicontinuity and nets	p. 563
Operations on correspondences	p. 566
The Maximum Theorem	p. 569
Vector-valued correspondences	p. 571
Demicontinuous correspondences	p. 574
Knaster-Kuratowski-Mazurkiewicz mappings	p. 577
Fixed point theorems	p. 581
Contraction correspondences	p. 585
Continuous selectors	p. 587
Measurable correspondences	p. 591
Measurability notions	p. 592
Compact-valued correspondences as functions	p. 597
Measurable selectors	p. 600
Correspondences with measurable graph	p. 606
Correspondences with compact convex values	p. 609
Integration of correspondences	p. 614
Markov transitions	p. 621
Markov and stochastic operators	p. 623
Markov transitions and kernels	p. 625
Continuous Markov transitions	p. 631
Invariant measures	p. 631
Ergodic measures	p. 636
Markov transition correspondences	p. 638
Random functions	p. 641
Dilations	p. 645
More on Markov operators	p. 650
A note on dynamical systems	p. 652
Ergodicity	p. 655
Measure-preserving transformations and ergodicity	p. 656
Birkhoff's Ergodic Theorem	p. 659
Ergodic operators	p. 661
References	p. 667
Index	p. 681
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Infinite Dimensional Analysis

A Hitchhiker's Guide

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Paperback

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