Elementary Functional Analysis (eBook)

Barbara MacCluer (Autor)

eBook Download: PDF

2008 | 1. Auflage
212 Seiten
Springer New York (Verlag)
978-0-387-85529-5 (ISBN)

Infos

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).

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Functional analysis arose in the early twentieth century and gradually, conquering one stronghold after another, became a nearly universal mathematical doctrine, not merely a new area of mathematics, but a new mathematical world view. Its appearance was the inevitable consequence of the evolution of all of nineteenth-century mathematics, in particular classical analysis and mathematical physics. Its original basis was formed by Cantor's theory of sets and linear algebra. Its existence answered the question of how to state general principles of a broadly interpreted analysis in a way suitable for the most diverse situations. A.M. Vershik ([45], p. 438). This text evolved from the content of a one semester introductory course in fu- tional analysis that I have taught a number of times since 1996 at the University of Virginia. My students have included ?rst and second year graduate students prep- ing for thesis work in analysis, algebra, or topology, graduate students in various departments in the School of Engineering and Applied Science, and several und- graduate mathematics or physics majors. After a ?rst draft of the manuscript was completed, it was also used for an independent reading course for several und- graduates preparing for graduate school.

Inhalt

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

Produkt-Details

• Erscheint lt. Verlag: 1.1.2009
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Analysis
• Themenwelt: Technik
• ISBN-10: 0-387-85529-7 / 0387855297
• ISBN-13: 978-0-387-85529-5 / 9780387855295