Subject(s) 
Banach spaces


Banach algebras

Physical Description 
viii, 371 p. ; 24 cm 
Note 
Includes bibliographical references (p. [354]362) and indexes 
Abstract 
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 specialize 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 BanachAlaoglu, HahnBanach, 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 
Contents 
pt. I. Introduction to Banach spaces. 1. Preliminaries  2. Elements of normed spaces  3. Banach spaces  pt. II. Introduction to Banach algebras. 4. Banach algebras  5. Representation theory  6. Algebras with an involution  7. The Borel functional calculus  pt. III. Several complex variables and Banach algebras. 8. Introduction to several complex variables  9. The holomorphic functional calculus in several variables 
Series 
Oxford graduate texts in mathematics ; 20

Alternate Author 
Dales, H. G. (Harold G.), 1944

