# Book:Joseph Muscat/Functional Analysis: An Introduction to Metric Spaces, Hilbert Spaces, and Banach Algebras

## Joseph Muscat: Functional Analysis: An Introduction to Metric Spaces, Hilbert Spaces, and Banach Algebras

Published $\text {2014}$, Springer

ISBN 978-3319067278.

### Subject Matter

Functional Analysis

### Contents

Introduction

Part I: Metric Spaces
2. Distance
2.1 Balls and Open Sets
2.2 Closed Sets
3. Convergence and Continuity
3.1 Convergence
3.2 Continuity
4. Completeness and Separability
4.1 Completeness
4.2 Uniformly Continuous Maps
4.3 Separable Spaces
5. Connectedness
5.1 Connected Sets
5.2 Components
6. Compactness
6.1 Bounded Sets
6.2 Totally Bounded Sets
6.3 Compact Sets
6.4 The Space $\map C {X, Y}$
Part II: Banach and Hilbert Spaces
7. Normed Spaces
7.1 Vector Spaces
7.2 Norms
7.3 Metric and Vector Properties
7.4 Complete and Separable Normed Vector Spaces
7.5 Series
8. Continuous Linear Maps
8.1 Operators
8.2 Quotient Spaces
8.3 $\R^N$ and Totally Bounded Sets
9. Main Examples
9.1 Sequence Spaces
9.2 Functions Spaces
10. Hilbert Spaces
10.1 Inner Products
10.2 Least Squares Approximation
10.3 Duality $H^* \approx H$
10.4 The Adjoint Map $T^*$
10.5 Inverse Problems
10.6 Orthonormal Bases
11. Banach Spaces
11.1 The Open Mapping Theorem
11.2 Compact Operators
11.3 The Dual Space $X^*$
11.4 The Adjoint $T^T$
11.5 Pointwise and Weak Convergence
12. Differentiation and Integration
12.1 Differentiation
12.2 Integration for Vector-Valued Functions
12.3 Complex Differentiation and Integration
Part III: Banach Algebras
13. Banach Algebras
13.1 Introduction
13.2 Power Series
13.3 The Group of Invertible Elements
13.4 Analytic Functions
14. Spectral Theory
14.2 The Spectrum of an Operator
14.3 Spectra of Compact Operators
14.4 The Functional Calculus
14.5 The Gelfand Transform
15. $C^*$-Algebras
15.1 Normal Elements
15.2 Norml Operators in $\map B H$
15.3 The Spectral Theorem for Compact Normal Operators
15.4 Representation Theorems

Hints to Selected Problems

Glossary of Symbols