Muscat J. Functional Analysis: An Introduction to Metric Spaces, Hilbert Spaces, and Banach Algebras

2nd edition. — Springer, 2024. — 462 p.This textbook provides an introduction to functional analysis suitable for lecture courses to final year undergraduates or beginning graduates. Starting from the very basics of metric spaces, the book adopts a self-contained approach to Banach spaces and operator theory that covers the main topics, including the spectral theorem, the Gelfand transform, and Banach algebras. Various applications, such as least squares approximation, inverse problems, and Tikhonov regularization, illustrate the theory. Over 1000 worked examples and exercises of varying difficulty present the reader with ample material for reflection. This new edition of Functional Analysis has been completely revised and corrected, with many passages rewritten for clarity, numerous arguments simplified, and a good amount of new material added, including new examples and exercises. The prerequisites, however, remain the same with only knowledge of linear algebra and real analysis of a single variable assumed of the reader.Preface
Aim of This Book
Audience
Acknowledgments
Introduction
Preliminaries
Metric Spaces
Distance
Convergence and Continuity
Completeness and Separability
Connectedness
Compactness
Banach and Hilbert Spaces
Normed Spaces
Continuous Linear Maps
The Classical Spaces
Hilbert Spaces
Banach Spaces
Differentiation and Integration
Banach Algebras
Banach Algebras
Spectral Theory
C*-Algebras
Hints to Selected Problems
Glossary of Symbols
Further Reading
More Advanced Books
Other References
Selected Historical Articles
Index