A Guide to Functional Analysis (Dolciani Mathematical Expositions, Series Number 49)

This book is a quick but precise and careful introduction to the subject of functional analysis. It covers the basic topics that can be found in a basic graduate analysis text. But it also covers more sophisticated topics such as spectral theory, convexity, and fixed-point theorems. A special feature of the book is that it contains a great many examples and even some applications. It concludes with a statement and proof of Lomonosov's dramatic result about invariant subspaces Cover 1 Title 4 Copyright 3 Contents 10 Preface 12 Chapter 1. Fundamentals 14 1.1 What is Functional Analysis? 14 1.2 Normed Linear Spaces 15 1.3 Finite-Dimensional Spaces 18 1.4 Linear Operators 19 1.5 The Baire Category Theorem 21 1.6 The Three Big Results 22 1.7 Applications of the Big Three 28 Chapter 2. Ode to the Dual Space 40 2.1 Introduction 40 2.2 Consequences of the Hahn-Banach Theorem 42 Chapter 3. Hilbert Space 46 3.1 Introduction 46 3.2 The Geometry of Hilbert Space 49 Chapter 4. The Algebra of Operators 58 4.1 Preliminaries 58 4.2 The Algebra of Bounded Linear Operators 60 4.3 Compact Operators 63 Chapter 5. Banach Algebra Basics 72 5.1 Introduction to Banach Algebras 72 5.2 The Structure of a Banach Algebra 76 5.3 Ideals 79 5.4 The Wiener Tauberian Theorem 85 Chapter 6. Topological Vector Spaces 88 6.1 Basic Ideas 88 6.2 Frechet Spaces 91 Chapter 7. Distributions 94 7.1 Preliminary Remarks 94 7.2 What is a Distribution? 95 7.3 Operations on Distributions 96 7.4 Approximation of Distributions 98 7.5 The Fourier Transform 100 Chapter 8. Spectral Theory 102 8.1 Background 102 8.2 The Main Result 104 Chapter 9. Convexity 112 9.1 Introductory Thoughts 112 9.2 Separation Theorems 113 9.3 The Main Result 116 Chapter 10. Fixed-Point Theorems 118 10.1 Banach’s Theorem 118 10.2 Two Applications 121 10.3 The Schauder Theorem 125 Table of Notation 128 Glossary 132 Bibliography 142 Index 146 About the Author 150 "functional analysis,analysis" functional analysis The purpose of A Guide to Functional Analysis is to introduce the reader with minimal background to the basic scripture of functional analysis. Readers should know some real analysis and some linear algebra. Measure theory rears its ugly head in some of the examples and also in the treatment of spectral theory. The latter is unavoidable and the former allows us to present a rich variety of examples. The nervous reader may safely skip any of the measure theory and still derive a lot from the rest of the book. Apart from this caveat, the book is almost completely self-contained; in a few instances we mention easily accessible references. A feature that sets this book apart from most other functional analysis texts is that it has a lot of examples and a lot of applications. This helps to make the material more concrete, and relates it to ideas that the reader has already seen. It also makes the book more accessible to a broader audience Functional analysis is an abstract and powerful modern theory that occupies a central role in mathematics. This book provides a quick but precise introduction to the subject, covering everything that a beginning graduate student needs to know. The subject has its roots in the theory of infinite-dimensional vector spaces which is where the book begins, with preliminaries from the theory of normed linear spaces, Hilbert spaces, operator algebras and distributions. The reader will then encounter more advanced topics such as spectral theory, convexity and fixed-point theorems. It contains plenty of examples and exercises, making it an ideal basis for advanced undergraduate and graduate level courses in a subject that has become an essential part of every analyst's toolkit. Fundamentals -- Ode To The Dual Space -- Hilbert Space -- The Algebra Of Operators -- Banach Algebra Basics -- Topological Vector Spaces -- Distributions -- Spectral Theory -- Convexity -- Fixed-point Theorems. Steven G. Krantz, Washington University In St. Louis. Maa Guides Series Numbering On Title Page Appears As # 49. It Should Read # 9. Includes Bibliographical References (pages 129-131) And Index.