Description
Book SynopsisBanach 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 int
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Table of ContentsPART I INTRODUCTION TO BANACH SPACES ; 1. Preliminaries ; 2. Elements of normed spaces ; 3. Banach spaces ; PART II BANACH ALGEBRAS ; 4. Banach algebras ; 5. Representation theory ; 6. Algebras with an involution ; 7. The Borel functional calculus ; PART III SCV AND BANACH ALGEBRAS ; 8. Introduction to several complex variables ; 9. The holomorphic functional calculus in several variables ; Bibliography ; Index
