Introduction to Banach Spaces and Algebras

by Allan, Graham; Dales, H. Garth

• ISBN13: 9780199206544
• ISBN10: 0199206546
• Format: Paperback
• Copyright: 2010-12-30
• Publisher: Oxford University Press

Summary

Banach spaces and algebras are a key topic of pure mathematics. Graham Allan's careful and detailed introductory account will prove essential reading for anyone wishing to specialise in functional analysis and is aimed at final year undergraduates or masters level students. Based on the author's lectures to fourth year students at Cambridge University, the book assumes knowledge typical of first degrees in mathematics, including metric spaces, analytic topology, and complex analysis. However, readers are not expected to be familiar with the Lebesgue theory of measure and integration. The text begins by giving the basic theory of Banach spaces, including dual spaces and bounded linear operators. It establishes forms of the theorems that are the pillars of functional analysis, including the Banach-Alaoglu, Hahn-Banach, uniform boundedness, open mapping, and closed graph theorems. There are applications to Fourier series and operators on Hilbert spaces. The main body of the text is an introduction to the theory of Banach algebras. A particular feature is the detailed account of the holomorphic functional calculus in one and several variables; all necessary background theory in one and several complex variables is fully explained, with many examples and applications considered. Throughout, exercises at sections ends help readers test their understanding, while extensive notes point to more advanced topics and sources. The book was edited for publication by Professor H. G. Dales of Leeds University, following the death of the author in August, 2007.

Author Biography

Graham Allan was a student of mathematics at Cambridge University, England, and obtained his PhD there in 1964. After periods as a Fellow at Cambridge, as a lecturer at the University of Newcastle-upon-Tyne, and as Professor of Pure Mathematics at the University of Leeds, Graham returned to Cambridge as a lecturer in mathematics in 1978, and he became a Reader in Functional Analysis in 1980. Graham was a Fellow of Churchill College from 1978, Director of Studies in Mathematics from 1985, and Vice-Master of the College 1990-93.

Table of Contents

Introduction	p. 1
Introduction to Banach Spaces
Preliminaries	p. 7
Remarks on set theory	p. 8
Metric spaces and analytic topology	p. 16
Complex analysis	p. 30
Elements of normed spaces	p. 33
Definitions and basic examples	p. 33
Weierstrass approximation theorems	p. 60
Inner-product spaces	p. 70
Elementary ideas on Fourier series	p. 84
Fourier integrals and Hermite functions	p. 98
Banach spaces	p. 106
Existence of continuous linear functionals	p. 106
Separation theorems	p. 122
Category theorems	p. 128
Dual operators	p. 140
Introduction to Banach Algebras
Banach algebras	p. 155
Elementary theory	p. 155
Commutative Banach algebras	p. 185
Runge's theorem and the holomorphic functional calculus	p. 211
Representation theory	p. 226
Representations and modules	p. 226
Automatic continuity	p. 241
Variation of the spectral radius	p. 248
Algebras with an involution	p. 260
Banach algebras with an involution	p. 260
C*-algebras	p. 269
The Borel functional calculus	p. 285
The Daniell integral	p. 285
The Borel functional calculus and the spectral theorem	p. 294
Several Complex Variables and Banach Algebras
Introduction to several complex variables	p. 305
Differentiable functions in the plane	p. 305
Functions of several variables	p. 313
Polynomial convexity	p. 326
The holomorphic functional calculus in several variables	p. 339
The main theorem	p. 339
Applications of the functional calculus	p. 345
References	p. 354
Index of terms	p. 363
Index of symbols	p. 369
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