This book introduces functional analysis to undergraduate mathematics students who possess a basic background in analysis and linear algebra. By studying how the Volterra operator acts on vector spaces of continuous functions, its readers will sharpen their skills, reinterpret what they already know, and learn fundamental Banach-space techniques—all in the pursuit of two celebrated results: the Titchmarsh Convolution Theorem and the Volterra Invariant Subspace Theorem. Exercises throughout the text enhance the material and facilitate interactive study.
Author(s): Joel H Shapiro
Series: Student Mathematical Library 85
Publisher: American Mathematical Society
Year: 2018
Language: English
Pages: 219
Table of Contents
Cover 1
Title page 4
Contents 8
Preface 12
List of Symbols 16
Part 1 . From Volterra to Banach 18
Chapter 1. Starting Out 20
1.1. A vector space 20
1.2. A linear transformation 21
1.3. Eigenvalues 23
1.4. Spectrum 25
1.5. Volterra spectrum 26
1.6. Volterra powers 28
1.7. Why justify our “formal calculation”? 30
1.8. Uniform convergence 31
1.9. Geometric series 33
Notes 36
Chapter 2. Springing Ahead 38
2.1. An initial-value problem 38
2.2. Thinking differently 41
2.3. Thinking linearly 42
2.4. Establishing norms 43
2.5. Convergence 45
2.6. Mass-spring revisited 49
2.7. Volterra-type integral equations 52
Notes 52
Chapter 3. Springing Higher 54
3.1. A general class of initial-value problems 54
3.2. Solving integral equations of Volterra type 56
3.3. Continuity in normed vector spaces 58
3.4. What’s the resolvent kernel? 62
3.5. Initial-value problems redux 66
Notes 68
Chapter 4. Operators as Points 70
Overview 70
4.1. How “big” is a linear transformation? 71
4.2. Bounded operators 73
4.3. Integral equations done right 78
4.4. Rendezvous with Riemann 80
4.5. Which functions are Riemann integrable? 84
4.6. Initial-value problems à la Riemann 86
Notes 90
Part 2 . Travels with Titchmarsh 96
Chapter 5. The Titchmarsh Convolution Theorem 98
5.1. Convolution operators 98
5.2. Null spaces 101
5.3. Convolution as multiplication 103
5.4. The One-Half Lemma 106
Notes 112
Chapter 6. Titchmarsh Finale 114
6.1. The Finite Laplace Transform 114
6.2. Stalking the One-Half Lemma 116
6.3. The complex exponential 120
6.4. Complex integrals 122
6.5. The (complex) Finite Laplace Transform 124
6.6. Entire functions 125
Notes 128
Part 3 . Invariance Through Duality 130
Chapter 7. Invariant Subspaces 132
7.1. Volterra-Invariant Subspaces 132
7.2. Why study invariant subspaces? 134
7.3. Consequences of the VIST 140
7.4. Deconstructing the VIST 143
Notes 148
Chapter 8. Digging into Duality 150
8.1. Strategy for proving \conjc 150
8.2. The “separable” Hahn-Banach Theorem 153
8.3. The “nonseparable” Hahn-Banach Theorem 161
Notes 166
Chapter 9. Rendezvous with Riesz 172
9.1. Beyond Riemann 172
9.2. From Riemann & Stieltjes to Riesz 177
9.3. Riesz with rigor 179
Notes 186
Chapter 10. V-Invariance: Finale 190
10.1. Introduction 190
10.2. One final reduction! 191
10.3. Toward the Proof of Conjecture U 192
10.4. Proof of Conjecture U 195
Notes 197
Appendix A. Uniform Convergence 200
Appendix B. \CComplex Primer 202
B.1. Complex numbers 202
B.2. Some Complex Calculus 204
B.3. Multiplication of complex series 205
B.4. Complex power series 207
Appendix C. Uniform Approximation by Polynomials 212
Appendix D. Riemann-Stieltjes Primer 216
Notes 228
Bibliography 230
Index 234
Back Cover 240
Preview Material
Preface
Table of Contents
