Book:Graham R. Allan/Introduction to Banach Spaces and Algebras

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Graham R. Allan and H. Garth Dales: Introduction to Banach Spaces and Algebras

Published $\text {2011}$, Oxford University Press

ISBN 978-0-19-920653-7


Subject Matter


Contents

Introduction
PART I INTRODUCTION TO BANACH SPACES
1. Preliminaries
Remarks on set theory
Metric spaces and analytic topology
Complex analysis
2. Elements of normed spaces
Definitions and basic examples
Weierstrass approximation theorems
Inner-product spaces
Elementary ideas on Fourier series
Fourier integrals and Hermite functions
3. Banach spaces
Existence of continuous linear functionals
Separation theorems
Category theorems
Dual operators
PART II INTRODUCTION TO BANACH ALGEBRAS
4. Banach algebras
Elementary theory
Commutative Banach algebras
Runge's theorem and the holomorphic functional calculus
5. Representation theory
Representations and modules
Automatic continuity
Variation of the spectral radius
6. Algebras with an involution
Banach algebras with an involution
$C^\ast$-algebras
7. The Borel functional calculus
The Daniell integral
The Borel functional calculus and the spectral theorem
PART III SEVERAL COMPLEX VARIABLES AND BANACH ALGEBRAS
8. Introduction to several complex variables
Differentiable functions in the plane
Functions of several complex variables
Polynomial convexity
9. The holomorphic functional calculus in several variables
The main theorem
Applications of the functional calculus
References
Index of terms
Index of symbols