Cover

Title page

Contents

Preface

List of Symbols

Part 1 . From Volterra to Banach

Chapter 1. Starting Out

1.1. A vector space

1.2. A linear transformation

1.3. Eigenvalues

1.4. Spectrum

1.5. Volterra spectrum

1.6. Volterra powers

1.7. Why justify our “formal calculation”?

1.8. Uniform convergence

1.9. Geometric series

Notes

Chapter 2. Springing Ahead

2.1. An initial-value problem

2.2. Thinking differently

2.3. Thinking linearly

2.4. Establishing norms

2.5. Convergence

2.6. Mass-spring revisited

2.7. Volterra-type integral equations

Notes

Chapter 3. Springing Higher

3.1. A general class of initial-value problems

3.2. Solving integral equations of Volterra type

3.3. Continuity in normed vector spaces

3.4. What’s the resolvent kernel?

3.5. Initial-value problems redux

Notes

Chapter 4. Operators as Points

Overview

4.1. How “big” is a linear transformation?

4.2. Bounded operators

4.3. Integral equations done right

4.4. Rendezvous with Riemann

4.5. Which functions are Riemann integrable?

4.6. Initial-value problems à la Riemann

Notes

Part 2 . Travels with Titchmarsh

Chapter 5. The Titchmarsh Convolution Theorem

5.1. Convolution operators

5.2. Null spaces

5.3. Convolution as multiplication

5.4. The One-Half Lemma

Notes

Chapter 6. Titchmarsh Finale

6.1. The Finite Laplace Transform

6.2. Stalking the One-Half Lemma

6.3. The complex exponential

6.4. Complex integrals

6.5. The (complex) Finite Laplace Transform

6.6. Entire functions

Notes

Part 3 . Invariance Through Duality

Chapter 7. Invariant Subspaces

7.1. Volterra-Invariant Subspaces

7.2. Why study invariant subspaces?

7.3. Consequences of the VIST

7.4. Deconstructing the VIST

Notes

Chapter 8. Digging into Duality

8.1. Strategy for proving \conjc

8.2. The “separable” Hahn-Banach Theorem

8.3. The “nonseparable” Hahn-Banach Theorem

Notes

Chapter 9. Rendezvous with Riesz

9.1. Beyond Riemann

9.2. From Riemann & Stieltjes to Riesz

9.3. Riesz with rigor

Notes

Chapter 10. V-Invariance: Finale

10.1. Introduction

10.2. One final reduction!

10.3. Toward the Proof of Conjecture U

10.4. Proof of Conjecture U

Notes

Appendix A. Uniform Convergence

Appendix B. \CComplex Primer

B.1. Complex numbers

B.2. Some Complex Calculus

B.3. Multiplication of complex series

B.4. Complex power series

Appendix C. Uniform Approximation by Polynomials

Appendix D. Riemann-Stieltjes Primer

Notes

Bibliography

Index

Back Cover