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Volterra adventures

Author: Joel H Shapiro
Publisher: Providence, Rhode Island : American Mathematical Society, [2018]
Series: Student mathematical library, v. 85.
Edition/Format:   eBook : Document : EnglishView all editions and formats
Summary:
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  Read more...
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Genre/Form: Electronic books
Additional Physical Format: Print version:
(DLC) 2017052435
(OCoLC)1013988578
Material Type: Document, Internet resource
Document Type: Internet Resource, Computer File
All Authors / Contributors: Joel H Shapiro
ISBN: 9781470447335 1470447339
OCLC Number: 1043211199
Description: 1 online resource
Contents: 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 problems3.2. 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 spaces5.3. 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 Theorem8.3. 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 PolynomialsAppendix D. Riemann-Stieltjes Primer; Notes; Bibliography; Index; Back Cover.
Series Title: Student mathematical library, v. 85.
Responsibility: Joel H. Shapiro.

Abstract:

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  Read more...

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