Infinite Dimensional Analysis: A Hitchhiker's Guide

by Aliprantis, Charalambos D.; Border, Kim C.

• ISBN13: 9783540658542
• ISBN10: 3540658548
• Edition: 2nd
• Format: Paperback
• Copyright: 1999-08-01
• Publisher: Springer Verlag

Summary

This book is intended for the student or researcher who could benefit from functional analytic methods, but who does not have an extensive background and does not plan to make a career as a functional analyst. It develops topology, convexity, Banach lattices, integration, correspondences (multifunctions), and the analytic approach to Markov processes. Many of the results were previously available only in esoteric monographs. The choice of material was motivated from problems in control theory and economics, although the material is more applicable than applied.

Table of Contents

Preface to the second edition	vii
Preface to the first edition	ix
A foreword to the practical	xvii
Odds and ends	1	(18)
Set theoretic notation	1	(1)
Relations, correspondences, and functions	2	(2)
A bestiary of relations	4	(1)
Equivalence relations	5	(1)
Orders and such	6	(1)
Numbers	7	(1)
Real functions	8	(1)
Duality of evaluation	9	(1)
Infinities	9	(2)
The axiom of choice and axiomatic set theory	11	(2)
Zorn's Lemma	13	(3)
Ordinals	16	(3)
Topology	19	(50)
Topological spaces	21	(3)
Neighborhoods and closures	24	(2)
Dense subsets	26	(1)
Nets	27	(4)
Filters	31	(3)
Nets and Filters	34	(1)
Continuous functions	35	(2)
Compactness	37	(4)
Nets vs. sequences	41	(1)
Semicontinuous functions	42	(2)
Separation properties	44	(2)
Comparing topologies	46	(1)
Weak topologies	47	(3)
The product topology	50	(3)
Pointwise and uniform convergence	53	(2)
Locally compact spaces	55	(3)
The Stone---Cech compactification	58	(5)
Stone---Cech compactification of a discrete set	63	(3)
Paracompact spaces and partitions of unity	66	(3)
Metrizable spaces	69	(54)
Metric spaces	70	(3)
Completeness	73	(3)
Uniformly continuous functions	76	(2)
Distance functions	78	(1)
Embeddings and completions	79	(1)
Compactness and completeness	80	(4)
Countable products of metric spaces	84	(1)
The Hilbert cube and metrization	85	(3)
The Baire Category Theorem	88	(2)
Contraction mappings	90	(3)
The Cantor set	93	(3)
The Baire space NN	96	(6)
Uniformities	102	(2)
The Hausdorff distance	104	(3)
The Hausdorff metric topology	107	(7)
Topologies for spaces of subsets	114	(4)
The space C (X, Y)	118	(3)
Semicontinuous functions	121	(2)
Measurability	123	(38)
Algebras of sets	125	(2)
Rings and semirings of sets	127	(4)
Dynkin's lemma	131	(2)
The Borel σ-algebra	133	(3)
Measurable functions	136	(1)
The space of measurable functions	137	(4)
Simple functions	141	(3)
The σ-algebra induced by a function	144	(1)
Product structures	145	(5)
Caratheodory functions	150	(4)
Borel functions and continuity	154	(2)
The Baire σ-algebra	156	(5)
Topological vector spaces	161	(76)
Linear topologies	164	(2)
Absorbing and circled sets	166	(4)
Convex sets	170	(5)
Convex and concave functions	175	(3)
Convex functions on finite dimensional spaces	178	(2)
Sublinear functions and gauges	180	(4)
The Hahn---Banach Extension Theorem	184	(2)
Separating hyperplane theorems	186	(3)
Separation by continuous functionals	189	(2)
Locally convex spaces and seminorms	191	(2)
Separation in locally convex spaces	193	(2)
Finite dimensional topological vector spaces	195	(4)
Supporting hyperplanes and cones	199	(7)
Dual pairs	206	(2)
Topologies consistent with a given dual	208	(2)
Polars	210	(6)
G-topologies	216	(3)
The Mackey topology	219	(2)
More about support functionals	221	(5)
The strong topology	226	(1)
Extreme points	226	(5)
Polytopes and weak neighborhoods	231	(6)
Normed spaces	237	(26)
Normed and Banach spaces	239	(1)
Linear operators on normed spaces	240	(5)
The norm dual of a normed space	245	(1)
The uniform boundedness principle	246	(4)
Weak topologies on normed spaces	250	(2)
Metrizability of weak topologies	252	(5)
Spaces of convex sets	257	(1)
Continuity of the evaluation	258	(2)
Adjoint operators	260	(3)
Riesz spaces	263	(38)
Orders, lattices, and cones	264	(1)
Riesz spaces	265	(2)
Order bounded sets	267	(1)
Order and lattice properties	268	(4)
The Riesz decomposition property	272	(1)
Disjointness	272	(1)
Riesz subspaces and ideals	273	(1)
Order convergence and order continuity	274	(2)
Bands	276	(2)
Positive functionals	278	(5)
Extending positive functionals	283	(2)
Positive operators	285	(2)
Topological Riesz spaces	287	(5)
The band generated by E'	292	(2)
Riesz pairs	294	(2)
Symmetric Riesz pairs	296	(5)
Banach lattices	301	(30)
Frechet and Banach lattices	302	(4)
Lattice homomorphisms and isometries	306	(2)
Order continuous norms	308	(2)
AM- and AL-spaces	310	(5)
The interior of the positive cone	315	(2)
The curious AL-space BVo	317	(5)
The Stone---Weierstrass Theorem	322	(1)
Projections and the fixed space of an operator	323	(3)
The Bishop---Phelps Theorem	326	(5)
Charges and measures	331	(34)
Set functions	334	(5)
Limits of sequences of measures	339	(1)
Outer measures and measurable sets	340	(2)
The Caratheodory extension of a measure	342	(6)
Measure spaces	348	(2)
Lebesgue measure	350	(2)
Product measures	352	(1)
Measures on Rn	353	(3)
Atoms	356	(2)
The AL-space of charges	358	(2)
The AL-space of measures	360	(2)
Absolute continuity	362	(3)
Measures and topology	365	(30)
Borel measures and regularity	366	(4)
Regular Borel measures	370	(4)
The support of a measure	374	(2)
Nonatomic Borel measures	376	(3)
Analytic sets	379	(11)
The Choquet Capacity Theorem	390	(5)
Integrals	395	(32)
The integral of a step function	396	(3)
Finitely additive integration of bounded functions	399	(1)
The Lebesgue integral	400	(5)
Continuity properties of the Lebesgue integral	405	(4)
The extended Lebesgue integral	409	(2)
Iterated integrals	411	(1)
The Riemann integral	412	(3)
The Bochner integral	415	(7)
The Gel fand integral	422	(3)
The Dunford and Pettis integrals	425	(2)
Lp-spaces	427	(28)
Lp-norms	428	(1)
Inequalities of Holder and Minkowski	429	(3)
Dense subspaces of Lp-spaces	432	(1)
Sublattices of Lp-spaces	433	(1)
Separable L1-spaces and measures	434	(2)
The Radon---Nikodym Theorem	436	(2)
Equivalent measures	438	(2)
Duals of Lp-spaces	440	(2)
Lyapunov's Convexity Theorem	442	(4)
Convergence in measure	446	(2)
Convergence in measure in Lp-spaces	448	(3)
Change of variables	451	(4)
Riesz Representation Theorems	455	(18)
The AM-space Bb(Σ) and its dual	456	(3)
The dual of Cb (X) for normal spaces	459	(6)
The dual of Cc (X) for locally compact spaces	465	(2)
Baire vs. Borel measures	467	(2)
Homomorphisms between C (X)-spaces	469	(4)
Probability measures on metrizable spaces	473	(20)
The weak* topology on P (X)	474	(6)
Embedding X in P (X)	480	(2)
Properties of P (X)	482	(4)
The many faces of P (X)	486	(1)
Compactness in P (X)	487	(2)
The Kolmogorov Extension Theorem	489	(4)
Spaces of sequences	493	(30)
The basic sequence spaces	493	(2)
The sequence spaces RN and ϕ	495	(2)
The sequence space co	497	(2)
The sequence space c	499	(2)
The lp-spaces	501	(4)
l1 and the symmetric Riesz pair (l∞, l1)	505	(1)
The sequence space l∞	506	(5)
More on l'∞ = ba (N)	511	(3)
Embedding sequence spaces	514	(4)
Banach---Mazur limits and invariant measures	518	(3)
Sequences of vector spaces	521	(2)
Correspondences	523	(34)
Basic definitions	524	(2)
Continuity of correspondences	526	(6)
Hemicontinuity and nets	532	(2)
Operations on correspondences	534	(3)
The Maximum Theorem	537	(3)
Vector-valued correspondences	540	(3)
Demicontinuous correspondences	543	(2)
Knaster---Kuratowski---Mazurkiewicz mappings	545	(3)
Fixed point theorems	548	(4)
Contraction correspondences	552	(1)
Continuous selectors	553	(4)
Measurable correspondences	557	(30)
Measurability notions	558	(5)
Compact-valued correspondences as functions	563	(3)
Measurable selectors	566	(5)
Correspondences with measurable graph	571	(3)
Correspondences with compact convex values	574	(6)
Integration of correspondences	580	(7)
Markov transitions	587	(34)
Markov and stochastic operators	589	(3)
Markov transitions and kernels	592	(5)
Continuous Markov transitions	597	(1)
Invariant measures	598	(4)
Ergodic measures	602	(3)
Markov transition correspondences	605	(3)
Random functions	608	(5)
Dilations	613	(5)
More on Markov operators	618	(2)
A note on dynamical systems	620	(1)
Ergodicity	621	(14)
Measure-preserving transformations and ergodicity	622	(3)
Birkhoff's Ergodic Theorem	625	(2)
Ergodic operators	627	(8)
References	635	(16)
Index	651