Functional Analysis: An Introduction to Metric Spaces, Hilbert Spaces, and Banach Algebras, 2nd Edition

Joseph Muscat, University of Malta


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ISBN-13 (Print): 978-3-031-27536-4
ISBN-13 (Online): 978-3-031-27537-1
  • Provides a self-contained introduction to functional analysis, assuming only real analysis and linear algebra
  • Presents the essential aspects of metric spaces, Hilbert spaces, Banach spaces and Banach algebras
  • Includes interesting applications of Hilbert spaces such as least squares approximation and inverse problems
  • Prepares the reader for graduate-level mathematical analysis

This textbook is an introduction to functional analysis suited to final year undergraduates or beginning graduates. Its various applications of Hilbert spaces, including least squares approximation, inverse problems, and Tikhonov regularization, should appeal not only to mathematicians interested in applications, but also to researchers in related fields.

Functional Analysis adopts a self-contained approach to Banach spaces and operator theory that covers the main topics, based upon the classical sequence and function spaces and their operators. It assumes only a minimum of knowledge in elementary linear algebra and real analysis; the latter is redone in the light of metric spaces. It contains more than a thousand worked examples and exercises, which make up the main body of the book.

Book review of the first edition by Prof. Mark Hunacek for the Mathematical Association of America (MAA).

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Table of Contents

  1. Introduction: Preliminaries
    Part I Metric Spaces
  2. Distance: Balls and Open Sets; Closed Sets
  3. Convergence and Continuity: Convergence; Continuity
  4. Completeness and Separability: Completeness; Uniformly Continuous Maps; Separable Spaces
  5. Connectedness: Connected Sets; Components
  6. Compactness: Bounded Sets; Totally Bounded Sets; Compact Sets; The Space \(\small C(X,Y)\)
    Part II Banach and Hilbert Spaces
  7. Normed Spaces: Vector Spaces; Norms; Metric and Vector Properties; Complete and Separable Normed Vector Spaces; Series
  8. Continuous Linear Maps: Operators; Operator Norms; Isomorphisma and Projections; Quotient Spaces; \(\small\mathbb{R}^N\) and Totally Bounded Sets
  9. The Classical Spaces: Sequence Spaces; Function Spaces
  10. Hilbert Spaces: Inner Products; Least Squares Approximation; Duality \(\small H^*\approx H\); Inverse Problems; Orthonormal Bases
  11. Banach Spaces: The Open Mapping Theorem; Compact Operators; The Dual Space \(\small X^*\); The Adjoint \(\small T^\top\); Strong and Weak Convergence
  12. Differentiation and Integration: Differentiation; Integration for Vector-Valued Functions; Complex Differentiation and Integration
    Part III Banach Algebras
  13. Banach Algebras: Introduction; Power Series; The Group of Invertible Elements; Analytic Functions
  14. Spectral Theory: The Spectrum of \(\small T\); The Spectrum of an Operator; Spectra of Compact Operators; The Functional Calculus; The Gelfand Transform
  15. C*-Algebras: Normal Elements; Normal Operators in \(\small B(H)\); The Numerical Range; The Spectral Theorem for Compact Normal Operators; Ideals of Compact Operators; Representation Theorems; Positive Self-Adjoint Elements; Spectral Theorem for Normal Operators

    16. Hints to Selected Problems
    Glossary of Symbols
    Further Reading
    Index

Link to first edition.

List of Errata