An Introduction to Functional Analysis

ISBN-13: 9780521728393 / Angielski / Miękka / 2020 / 416 str.

James C. Robinson

Opis

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

Recenzje

'… 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

Spis treści

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.