Functional analysis. (English) Zbl 1009.47001

Pure and Applied Mathematics. A Wiley-Interscience Series of Texts, Monographs and Tracts. Chichester: Wiley (ISBN 0-471-55604-1/hbk). xx, 580 p. (2002).
There are several very good textbooks on functional analysis on the market, and the book under reviews figures well among them. The topics are carefully selected from the broad field of functional analysis, providing students with the necessary knowledge to start further research not only in functional analysis but also in related fields such as partial differential equations or mathematical physics. The selected topics cover the theory of Banach spaces, Banach algebras, spectral theory, semigroups and scattering theory. Interesting and helpful exercises are posted at the right places, and brief historical remarks supplement the text. The geometry of Banach spaces and aspects of non-linear functional analysis are not treated here.
The book is divided into the following chapters.
1. Linear Spaces, 2. Linear Maps, 3. The Hahn-Banach Theorem, 4. Applications of the Hahn-Banach Theorem, 5. Normed Linear Spaces, 6. Hilbert Spaces, 7. Applications of Hilbert Space Results, 8. Duals of Normed Spaces, 9. Applications of Duality, 10. Weak Convergence, 11. Applications of Weak Convergence, 12. The Weak and Weak\(^*\) Topologies, 13. Locally Convex Topologies and the Krein-Milman Theorem, 14. Examples of Convex Sets and Their Extreme Points, 15. Bounded Linear Maps, 16. Examples of Bounded Linear Maps, 17. Banach Algebras and their Elementary Spectral Theory, 18. Gelfand’s Theory of Commutative Banach Algebras, 19. Applications of Gelfand’s Theory of Commutative Banach Algebras, 20. Examples of Operators and Their Spectra, 21. Compact Maps, 22. Examples of Compact Operators, 23. Positive compact operators, 24. Fredholm’s Theory of Integral Equations, 25. Invariant Subspaces, 26. Harmonic Analysis on a Halfline, 27. Index Theory, 28. Compact Symmetric Operators in Hilbert Space, 29. Examples of Compact Symmetric Operators, 30. Trace Class and Trace Formula, 31. Spectral Theory of Symmetric, Normal, and Unitary Operators, 32. Spectral Theory of Self-Adjoint Operators, 33. Examples of Self-Adjoint Operators, 34. Semigroups of Operators, 35. Groups of Unitary Operators, 36. Examples of Strongly Continuous Semigroups, 37. Scattering Theory, 38. Theorem of Beurling. Appendix: A. Riesz-Kakutani Representation Theorem, B: Theory of Distributions, C: Zorn’s Lemma.
The book is highly recommended to all students of analysis.


47-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to operator theory
46-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to functional analysis