Maccluer Elementary Functional Analysis

1. Auflage 2008
ISBN: 978-0-387-85529-5
Verlag: Springer
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band 253, 212 Seiten

Reihe: Graduate Texts in Mathematics

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 authors 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 authors belief that some understanding of the historical context of the development of any field in mathematics both deepens and enlivens ones 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. 

The Author

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).

Autoren/Hrsg.

Maccluer, Barbara

Inhaltsverzeichnis

1;Preface;7
2;Contents;9
3;Hilbert Space Preliminaries;11
3.1;1.1 Normed Linear Spaces;12
3.2;1.2 Orthogonality;20
3.3;1.3 Hilbert Space Geometry;22
3.4;1.4 Linear Functionals;25
3.5;1.5 Orthonormal Bases;29
3.6;1.6 Exercises;33
4;Operator Theory Basics;40
4.1;2.1 Bounded Linear Operators;40
4.2;2.2 Adjoints of Hilbert Space Operators;43
4.3;2.3 Adjoints of Banach Space Operators;50
4.4;2.4 Exercises;52
5;The Big Three;57
5.1;3.1 The HahnÒBanach Theorem;58
5.2;3.2 Principle of Uniform Boundedness;63
5.3;3.3 Open Mapping and Closed Graph Theorems;69
5.4;3.4 Quotient Spaces;74
5.5;3.5 Banach and the Scottish Caf ï e;75
5.6;3.6 Exercises;76
6;Compact Operators;85
6.1;4.1 Finite-Dimensional Spaces;85
6.2;4.2 Compact Operators;88
6.3;4.3 A Preliminary Spectral Theorem;95
6.4;4.4 The Invariant Subspace Problem;102
6.5;4.5 Introduction to the Spectrum;104
6.6;4.6 The Fredholm Alternative;107
6.7;4.7 Exercises;109
7;Banach and C*- Algebras;114
7.1;5.1 First Examples;115
7.2;5.2 Results on Spectra;117
7.3;5.3 Ideals and Homomorphisms;127
7.4;5.4 Commutative Banach Algebras;131
7.5;5.5 Weak Topologies;134
7.6;5.6 The Gelfand Transform;139
7.7;5.7 The Continuous Functional Calculus;147
7.8;5.8 Fredholm Operators;150
7.9;5.9 Exercises;153
8;The Spectral Theorem;163
8.1;6.1 Normal Operators Are Multiplication Operators;163
8.2;6.2 Spectral Measures;171
8.3;6.3 Exercises;189
9;Real Analysis Topics;193
9.1;A.1 Measures;193
9.2;A.2 Integration;196
9.3;A.3 Lp Spaces;202
9.4;A.4 The StoneÒWeierstrass Theorem;203
9.5;A.5 Positive Linear Functionals on C( X);204
10;References;206
11;Index;208