Beginning Functional Analysis

Description

The unifying approach of functional analysis is to view functions as points in some abstract vector space and the differential and integral operators relating these points as linear transformations on these spaces. The author presents the basics of functional analysis with attention paid to both expository style and technical detail, while getting to interesting results as quickly as possible. The book is accessible to students who have completed first courses in linear algebra and real analysis. Topics are developed in their historical context, with accounts of the past - including biographies - appearing throughout the text. The book offers suggestions and references for further study, and many exercises. Karen Saxe is Associate Professor of Mathematics at Macalester College in St. Paul, Minnesota. She received her Ph.D. from the University of Oregon. Before joining the faculty at Macalester, she held a two-year FIPSE post-doctoral position at St. Olaf College in Northfield, Minnesota. She currently serves on the editorial board of the MAA's College Mathematics Journal. This is her first book. Beginning Functional Analysis......Page 1 Preface......Page 3 Introduction: To the Student......Page 11 1 Metric Spaces, Norrned Spaces, Inner Product Spaces......Page 13 1.1 Basic Definitions and Theorems......Page 14 1.2 Examples: Sequence Spaces and Function Spaces......Page 18 1.3 A Discussion About Dimension......Page 21 Exercises for Chapter 1......Page 23 2.1 Open, Closed, and Compact Sets; the Heine-Bore1 and Ascoli-Arzel; Theorems......Page 26 2.2 Separability......Page 32 2.3 Completeness: Banach and Hilbert Spaces......Page 33 Exercises for Chapter 2......Page 39 3 Measure and Integration......Page 42 3.1 Probability Theory as Motivation......Page 43 3.2 Lebesgue Measure on Euclidean Space......Page 45 3.3 Measurable and Lebesgue Integrable Functions on Euclidean Space......Page 54 3.4 The Convergence Theorems......Page 60 3.5 Comparison of the Lebesgue Integral with the Riemann Integral......Page 64 3.6 General Measures and the Lebesgue Lp-spaces: The Importance of Lebesgue's Ideas in Functional Analysis......Page 67 Exercises for Chapter 3......Page 79 4.1 Orthonormal Sequences......Page 84 4.2 Bessel's Inequality, Parseval's Theorem, and the Riesz-Fischer Theorem......Page 91 4.3 A Return to Classical Fourier Analysis......Page 95 Exercises for Chapter 4......Page 98 5.1 Basic Definitions and Examples......Page 102 5.2 Boundedness and Operator Norms......Page 105 5.3 Banach Algebras and Spectra; Compact Operators......Page 109 5.4 An Introduction to the Invariant Subspace Problem......Page 122 5.5 The Spectral Theorem for Compact Hermitian Operators......Page 134 Exercises for Chapter 5......Page 139 6 Further Topics......Page 145 6.1 The Classical Weierstrass Approximation Theorem and the Generalized Stone-Weierstrass Theorem......Page 146 6.2 The Baire Category Theorem with an Application to Real Analysis......Page 156 6.3 Three Classical Theorems from Functional Analysis......Page 161 6.4 The Existence of a Nonmeasurable Set......Page 168 6.5 Contraction Mappings......Page 170 6.6 The Function Space C([a, b]) as a Ring, and its Maximal Ideals......Page 174 6.7 Hilbert Space Methods in Quantum Mechanics......Page 177 Exercises for Chapter 6......Page 186 Appendix A Complex Numbers......Page 191 Exercises for Appendix A......Page 193 Appendix B Basic Set Theory......Page 195 Exercises for Appendix B......Page 196 References......Page 197 Index......Page 203 Undergraduate Texts in Mathematics......Page 208