Volterra Adventures (Student Mathematical Library) (Student Mathematical Library, 85)

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. Cover......Page 1 Title page......Page 4 Contents......Page 8 Preface......Page 12 List of Symbols......Page 16 Part 1 . From Volterra to Banach......Page 18 1.1. A vector space......Page 20 1.2. A linear transformation......Page 21 1.3. Eigenvalues......Page 23 1.4. Spectrum......Page 25 1.5. Volterra spectrum......Page 26 1.6. Volterra powers......Page 28 1.7. Why justify our “formal calculation”?......Page 30 1.8. Uniform convergence......Page 31 1.9. Geometric series......Page 33 Notes......Page 36 2.1. An initial-value problem......Page 38 2.2. Thinking differently......Page 41 2.3. Thinking linearly......Page 42 2.4. Establishing norms......Page 43 2.5. Convergence......Page 45 2.6. Mass-spring revisited......Page 49 Notes......Page 52 3.1. A general class of initial-value problems......Page 54 3.2. Solving integral equations of Volterra type......Page 56 3.3. Continuity in normed vector spaces......Page 58 3.4. What’s the resolvent kernel?......Page 62 3.5. Initial-value problems redux......Page 66 Notes......Page 68 Overview......Page 70 4.1. How “big” is a linear transformation?......Page 71 4.2. Bounded operators......Page 73 4.3. Integral equations done right......Page 78 4.4. Rendezvous with Riemann......Page 80 4.5. Which functions are Riemann integrable?......Page 84 4.6. Initial-value problems à la Riemann......Page 86 Notes......Page 90 Part 2 . Travels with Titchmarsh......Page 96 5.1. Convolution operators......Page 98 5.2. Null spaces......Page 101 5.3. Convolution as multiplication......Page 103 5.4. The One-Half Lemma......Page 106 Notes......Page 112 6.1. The Finite Laplace Transform......Page 114 6.2. Stalking the One-Half Lemma......Page 116 6.3. The complex exponential......Page 120 6.4. Complex integrals......Page 122 6.5. The (complex) Finite Laplace Transform......Page 124 6.6. Entire functions......Page 125 Notes......Page 128 Part 3 . Invariance Through Duality......Page 130 7.1. Volterra-Invariant Subspaces......Page 132 7.2. Why study invariant subspaces?......Page 134 7.3. Consequences of the VIST......Page 140 7.4. Deconstructing the VIST......Page 143 Notes......Page 148 8.1. Strategy for proving \conjc......Page 150 8.2. The “separable” Hahn-Banach Theorem......Page 153 8.3. The “nonseparable” Hahn-Banach Theorem......Page 161 Notes......Page 166 9.1. Beyond Riemann......Page 172 9.2. From Riemann & Stieltjes to Riesz......Page 177 9.3. Riesz with rigor......Page 179 Notes......Page 186 10.1. Introduction......Page 190 10.2. One final reduction!......Page 191 10.3. Toward the Proof of Conjecture U......Page 192 10.4. Proof of Conjecture U......Page 195 Notes......Page 197 Appendix A. Uniform Convergence......Page 200 B.1. Complex numbers......Page 202 B.2. Some Complex Calculus......Page 204 B.3. Multiplication of complex series......Page 205 B.4. Complex power series......Page 207 Appendix C. Uniform Approximation by Polynomials......Page 212 Appendix D. Riemann-Stieltjes Primer......Page 216 Notes......Page 228 Bibliography......Page 230 Index......Page 234 Back Cover......Page 240 "This book introduces functional analysis to undergraduate mathematics students who possess an 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."--Page 4 of cover