Infinite Dimensional Analysis: A Hitchhiker's Guide

by Charalambos D. Aliprantis

Odds and ends.- Set6 theoretic notation.- Relations, correspondences, and functions.- A bestiary of relations.- Equivalence relations.- Orders and such.- Numbers.- Real functions.- Duality of evaluation.- infinityies.- The axiom of choice and axiomatic set theory.- Zorn`s Lemma.- Ordinals.- Topology.- Topological spaces.- Neighborhoods and closures.- Dense subsets.- Nets.- Filters.- Nets and Filters.- Continuous functions.- Compactness.- Nets vs. sequences.- Semicontinuous functions.- Separation properties.- Comparing topologies.- Weak topologies.- The product topology.- Pointwise and uniform convergence.- Locally compact spaces.- The Stone-Cech compactification.- Stone-Cech compactification of a discrete set.- Paracompact spaces and partitions of unity.- Metrizable spaces.- Metric spaces.- Completeness.- Uniformly continuous functions.- Distance functions.- Embeddings and competions.- Compactness and completeness.- Countable products of metric spaces.- The Hilbert cube and metrization.- The Baire Category Theorem.- Contraction mappings.- The Cantor set.- The Baire space NN.- Uniformities.- The Hausdorff distance.- The Hausdorff metric topology.- Topologies for spaces of subsets.- The space C(X,Y).- Semicontinuous functions.- Measurability.- Algebras of sets.- Rings and semirings of sets.- Dynkin`s lemma.- The Borel ?-algebra.- Measurable functions.- The space of measurable functions.- Simple functions.- The ?-algebra induced by a function.- Product structures.- Carathéodory functions.- Borel functions and continuity.- The Baire ?-algebra.- Topological vector spaces.- Linear topologies.- Absorbing and circled sets.- Convex sets.- Convex and concave functions.- Convex functions on finite dimensional spaces.- Sublinear functions and gauges.- The Hahn-Banach Extension Theorem.- Separating hyperplane theorems.- Separation by continuous functionals.- Locally convex spaces and seminorms.- Separation in locally convex spaces.-Finite dimensional topological vector spaces.- Supporting hyperplanes and cones.- Dual pairs.- Topologies consistent with a given dual.- Polars.- The Mackey topology.- More about support functionals.- The strong topology.- Extreme points.- Polytopes and weak neighborhoods.- Normed spaces.- Normed and Banach spaces.- Linear operators on normed spaces.- The norm dual of a normed space.- The uniform boundedness principle.- Weak topologies on normed spaces.- Metrizability of weak topologies.- Spaces of convex sets.- Continuity of the evaluation.- Adjoint operators.- Riesz spaces.- Orders, lattices, and cones.- Riesz spaces.- Order bounded sets.- Order and lattice properties.- The Riesz decomposition property.- Disjointness.- Riesz subspaces and ideals.- Order convergence and order continuity.- Bands.- Positive functionals.- Extending positive functionals.- Positive operators.- Topological Riesz spaces.- The band generated by E'.- Riesz pairs.- Sysmmetric Riesz pairs.- Banach lattices.- Fréchet and Banach lattices.- Lattice homomorphisms and isometries.- Order continuous norms.- AM- and Al-spaces.- The interior of the positive cone.- The curious Al-space BVo.- The Stone-Weierstrass Theorem.- Projections and the fixed space of an operator.- The Bishop-Phelps Theorem.- Charges and measures.- Set functions.- Limits of sequences of measures.- Outer measures and measurable sets.- The Carathéodory extension of a measure.- Measure spaces.- Lebesgue measure.- Product measures.- Measures on Rn.- Atoms.- The AL-space of charges.- The Al-space of measures.- Absolute continuity.- Measures and topology.- Borel measures and regularity.- Regular Borel measures.- The support of a measure.- Nonatomic Borel measures.- Analytic sets.- The Choquet Capacity Theorem.- Integrals.- The integral of a step function.- Finitely additive integration of bounded functions.- The Lebesgue integral.- Continuity properties of

Genres: Mathematics

698 pages, Paperback

First published January 1, 1994

Reviews

[Peter Frazier · Rating 5 out of 5]

This book is a totally cool introduction to functional analysis. The presentation is clear and easy to follow without being slow paced, and in particular I like the treatment of weak* convergence more than Billingsley's.

Before reading this book, I knew squat. Afterward, of course I still know very little, but it taught me enough to be able to make respectable progress on some problems in infinite dimensional vector spaces. So I am a content reader.