An Introduction to Functional Analysis

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.

Table of Contents:
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(Ω); 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.

About the Author :
James C. Robinson is a professor in the Mathematics Institute at the University of Warwick. He has been the recipient of a Royal Society University Research Fellowship and an Engineering and Physical Sciences Research Council (EPSRC) Leadership Fellowship. He has written six books in addition to his many publications in infinite-dimensional dynamical systems, dimension theory, and partial differential equations.

Review :
'This excellent introduction to functional analysis brings the reader at a gentle pace from a rudimentary acquaintance with analysis to a command of the subject sufficient, for example, to start a rigorous study of partial differential equations. The choice and order of topics are very well thought-out, and there is a fine balance between general results and concrete examples and applications.' Charles Fefferman, Princeton University, New Jersey 'An Introduction to Functional Analysis covers everything that one would expect to meet in an undergraduate course on this elegant area and more, including spectral theory, the category-based theorems and unbounded operators. With a well-written narrative and clear detailed proofs, together with plentiful examples and exercises, this is both an excellent course book and a valuable reference for those encountering functional analysis from across mathematics and science.' Kenneth Falconer, University of St Andrews, Scotland '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 is a valuable book. It is an accessible yet serious look at the subject, and anybody who has worked through it will be rewarded with a good understanding of functional analysis, and should be in a position to read more advanced books with profit.' Mark Hunacek, The Mathematical Gazette