An Introduction to Functional Analysis

Accessible text covering core functional analysis topics in Hilbert and Banach spaces, with detailed proofs and 200 fully-worked exercises.

James C. Robinson (Author)

9780521728393, Cambridge University Press

Paperback, published 12 March 2020

416 pages, 17 b/w illus. 215 exercises
22.7 x 15.3 x 2.2 cm, 0.6 kg

'This is a beautifully written book, containing a wealth of worked examples and exercises, covering the core of the theory of Banach and Hilbert spaces. The book will be of particular interest to those wishing to learn the basic functional analytic tools for the mathematical analysis of partial differential equations and the calculus of variations.' Endre Suli, University of Oxford

This accessible text covers key results in functional analysis that are essential for further study in the calculus of variations, analysis, dynamical systems, and the theory of partial differential equations. The treatment of Hilbert spaces covers the topics required to prove the Hilbert–Schmidt theorem, including orthonormal bases, the Riesz representation theorem, and the basics of spectral theory. The material on Banach spaces and their duals includes the Hahn–Banach theorem, the Krein–Milman theorem, and results based on the Baire category theorem, before culminating in a proof of sequential weak compactness in reflexive spaces. Arguments are presented in detail, and more than 200 fully-worked exercises are included to provide practice applying techniques and ideas beyond the major theorems. Familiarity with the basic theory of vector spaces and point-set topology is assumed, but knowledge of measure theory is not required, making this book ideal for upper undergraduate-level and beginning graduate-level courses.

Part I. Preliminaries: 1. Vector spaces and bases
2. Metric spaces
Part II. Normed Linear Spaces: 3. Norms and normed spaces
4. Complete normed spaces
5. Finite-dimensional normed spaces
6. Spaces of continuous functions
7. Completions and the Lebesgue spaces Lp(?)
Part III. Hilbert Spaces: 8. Hilbert spaces
9. Orthonormal sets and orthonormal bases for Hilbert spaces
10. Closest points and approximation
11. Linear maps between normed spaces
12. Dual spaces and the Riesz representation theorem
13. The Hilbert adjoint of a linear operator
14. The spectrum of a bounded linear operator
15. Compact linear operators
16. The Hilbert–Schmidt theorem
17. Application: Sturm–Liouville problems
Part IV. Banach Spaces: 18. Dual spaces of Banach spaces
19. The Hahn–Banach theorem
20. Some applications of the Hahn–Banach theorem
21. Convex subsets of Banach spaces
22. The principle of uniform boundedness
23. The open mapping, inverse mapping, and closed graph theorems
24. Spectral theory for compact operators
25. Unbounded operators on Hilbert spaces
26. Reflexive spaces
27. Weak and weak-* convergence
Appendix A. Zorn's lemma
Appendix B. Lebesgue integration
Appendix C. The Banach–Alaoglu theorem
Solutions to exercises
References
Index.

Subject Areas: Functional analysis & transforms [PBKF]