Elementary Functional Analysis

✍ Scribed by Barbara MacCluer

• Publisher: Springer
• Year: 2009
• Tongue: English
• Leaves: 220
• Series: Graduate Texts in Mathematics, Vol. 253
• Edition: 1st Edition. 2nd Printing. 2008
• Category: Library

✦ Synopsis

Large number of high-quality exercises
Concise and gentle approach with a unique set of topics
Historical details are spread throughout the book
This book has been thoroughly class-tested

This text is intended for a one-semester introductory course in functional analysis for graduate students and well-prepared advanced undergraduates in mathematics and related fields. It is also suitable for self-study, and could be used for an independent reading course for undergraduates preparing to start graduate school.

While this book is relatively short, the author has not sacrificed detail. Arguments are presented in full, and many examples are discussed, making the book ideal for the reader who may be learning the material on his or her own, without the benefit of a formal course or instructor. Each chapter concludes with an extensive collection of exercises.

The choice of topics presented represents not only the author's preferences, but also her desire to start with the basics and still travel a lively path through some significant parts of modern functional analysis. The text includes some historical commentary, reflecting the author's belief that some understanding of the historical context of the development of any field in mathematics both deepens and enlivens one's appreciation of the subject.

The prerequisites for this book include undergraduate courses in real analysis and linear algebra, and some acquaintance with the basic notions of point set topology. An Appendix provides an expository discussion of the more advanced real analysis prerequisites, which play a role primarily in later sections of the book.

Barbara MacCluer is Professor of Mathematics at University of Virginia. She also co-authored a book with Carl Cowen, Composition Operators on Spaces of Analytic Functions (CRC 1995).

Content Level » Graduate

Related subjects » Analysis

✦ Table of Contents

Cover

Front Matter

 Graduate Texts in Mathematics 253

 Elementary Functional Analysis

 Copyright

      Springer Science+Business Media, LLC 2009

      ISBN 978-0-387-85528-8

      e-ISBN 978-0-387-855295

      DOI: 10.1007/978-0-387-855295

      Library of Congress Control Number: 2008937759

 Dedicated To Tom, Josh, and David

 Preface

 Content

Chapter 1 Hilbert Space Preliminaries

 1.1 Normed Linear Spaces

 1.2 Orthogonality

 1.3 Hilbert Space Geometry

 1.4 Linear Functionals

 1.5 Orthonormal Bases

 1.6 Exercises

Chapter 2 Operator Theory Basics

 2.1 Bounded Linear Operators

 2.2 Adjoints of Hilbert Space Operators

 2.3 Adjoints of Banach Space Operators

 2.4 Exercises

Chapter 3 The Big Three

 3.1 The Hahn–Banach Theorem

 3.2 Principle of Uniform Boundedness

 3.3 Open Mapping and Closed Graph Theorems

 3.4 Quotient Spaces

 3.5 Banach and the Scottish Caf´e

 3.6 Exercises

Chapter 4 Compact Operators

 4.1 Finite-Dimensional Spaces

 4.2 Compact Operators

 4.3 A Preliminary Spectral Theorem

 4.4 The Invariant Subspace Problem

 4.5 Introduction to the Spectrum

 4.6 The Fredholm Alternative

 4.7 Exercises

Chapter 5 Banach and C*-Algebras

 5.1 First Examples

 5.2 Results on Spectra

 5.3 Ideals and Homomorphisms

 5.4 Commutative Banach Algebras

 5.5 Weak Topologies

 5.6 The Gelfand Transform

 5.7 The Continuous Functional Calculus

 5.8 Fredholm Operators

 5.9 Exercises

Chapter 6 The Spectral Theorem

 6.1 Normal Operators Are Multiplication Operators

 6.2 Spectral Measures

 6.3 Exercises

Back Matter

 Appendix A  Real Analysis Topics

      A.1 Measures

      A.2 Integration

      A.3 Lp Spaces

      A.4 The Stone–Weierstrass Theorem

      A.5 Positive Linear Functionals on C(X)

 References

 Index

Back Cover