A First Course in Functional Analysis

✍ Scribed by S. David Promislow

• Publisher: Wiley
• Year: 2008
• Tongue: English
• Leaves: 291
• Category: Library

✦ Synopsis

A concise introduction to the major concepts of functional analysis
Requiring only a preliminary knowledge of elementary linear algebra and real analysis, A First Course in Functional Analysis provides an introduction to the basic principles and practical applications of functional analysis. Key concepts are illustrated in a straightforward manner, which facilitates a complete and fundamental understanding of the topic.

This book is based on the author's own class-tested material and uses clear language to explain the major concepts of functional analysis, including Banach spaces, Hilbert spaces, topological vector spaces, as well as bounded linear functionals and operators. As opposed to simply presenting the proofs, the author outlines the logic behind the steps, demonstrates the development of arguments, and discusses how the concepts are connected to one another. Each chapter concludes with exercises ranging in difficulty, giving readers the opportunity to reinforce their comprehension of the discussed methods. An appendix provides a thorough introduction to measure and integration theory, and additional appendices address the background material on topics such as Zorn's lemma, the Stone-Weierstrass theorem, Tychonoff's theorem on product spaces, and the upper and lower limit points of sequences. References to various applications of functional analysis are also included throughout the book.

A First Course in Functional Analysis is an ideal text for upper-undergraduate and graduate-level courses in pure and applied mathematics, statistics, and engineering. It also serves as a valuable reference for practitioners across various disciplines, including the physical sciences, economics, and finance, who would like to expand their knowledge of functional analysis.

✦ Table of Contents

1. Linear Spaces and Operators
2. Normed Linear Spaces: The Basics
3. Major Banach Space Theorems
4. Hilbert Spaces
5. Hahn-Banach Theorem
6. Duality
7. Topological Vector Spaces
8. The Spectrum
9. Compact Operators
10. Application to Integral and Differential Equations
11. Spectral Theorem for Bounded, Self-Adjoint Operators
    Appendix A Zorn's Lemma
    Appendix B Stone-Weierstrass Theorem
    Appendix C Extended Real Numbers and Limit Points of Sequences
    Appendix D Measure and Integration
    Appendix E Tychonoff's Theorem