Functional Analysis

This textbook on functional analysis offers a short and concise introduction to the subject. The book is designed in such a way as to provide a smooth transition between elementary and advanced topics and its modular structure allows for an easy assimilation of the content. Starting from a dedicated chapter on the axiom of choice, subsequent chapters cover Hilbert spaces, linear operators, functionals and duality, Fourier series, Fourier transform, the fixed point theorem, Baire categories, the uniform bounded principle, the open mapping theorem, the closed graph theorem, the Hahn-Banach theorem, adjoint operators, weak topologies and reflexivity, operators in Hilbert spaces, spectral theory of operators in Hilbert spaces, and compactness. Each chapter ends with workable problems.

The book is suitable for graduate students, but also for advanced undergraduates, in mathematics and physics.

Contents:

List of Figures

Basic Notation

Choice Principles

Hilbert Spaces

Completeness, Completion and Dimension

Linear Operators

Functionals and Dual Spaces

Fourier Series

Fourier Transform

Fixed Point Theorem

Baire Category Theorem

Uniform Boundedness Principle

Open Mapping Theorem

Closed Graph Theorem

Hahn-Banach Theorem

The Adjoint Operator

Weak Topologies and Reflexivity

Operators in Hilbert Spaces

Spectral Theory of Operators on Hilbert Spaces

Compactness

Bibliography

Index