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

Subject Matter

 * Functional Analysis

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