Functional Analysis: (Cambridge Studies in Advanced Mathematics)

This comprehensive introduction to functional analysis covers both the abstract theory and applications to spectral theory, the theory of partial differential equations, and quantum mechanics. It starts with the basic results of the subject and progresses towards a treatment of several advanced topics not commonly found in functional analysis textbooks, including Fredholm theory, form methods, boundary value problems, semigroup theory, trace formulas, and a mathematical treatment of states and observables in quantum mechanics. The book is accessible to graduate students with basic knowledge of topology, real and complex analysis, and measure theory. With carefully written out proofs, more than 300 problems, and appendices covering the prerequisites, this self-contained volume can be used as a text for various courses at the graduate level and as a reference text for researchers in the field.

Table of Contents:
1. Banach spaces; 2. The classical Banach spaces; 3. Hilbert spaces; 4. Duality; 5. Bounded operators; 6. Spectral theory; 7. Compact operators; 8. Bounded operators on Hilbert spaces; 9. The spectral theorem for bounded normal operators; 10. The spectral theorem for unbounded normal operators; 11. Boundary value problems; 12. Forms; 13. Semigroups of linear operators; 14. Trace class operators; 15. States and observables; Appendix A. Zorn's lemma; Appendix B. Tensor products; Appendix C. Topological spaces; Appendix D. Metric spaces; Appendix E. Measure spaces; Appendix F. Integration; Appendix G. Notes; References; Index.

About the Author :
Jan van Neerven holds an Antoni van Leeuwenhoek professorship at Delft University of Technology. Author of four books and more than 100 peer-reviewed articles, he is a leading expert in functional analysis and operator theory and their applications in stochastic analysis and the theory of partial differential equations.

Review :
'One of its strengths is that it is a genuine textbook rather than a reference text. It is highly readable and pedagogical, giving a good level of detail in proofs, but staying concise and keeping its story clear rather than being encyclopedic. Another strength of the textbook is that it is well motivated by applications of functional analysis to other areas of mathematics, with a special emphasis on partial differential equations and quantum mechanics throughout the book.' Pierre Portal, zbMATH Open 'Everything is beautifully and clearly expressed. In short, highly recommended!' Klaas Landsman, Nieuw Archief voor Wiskunde