'Functional Analysis Revisited' is not a first course in functional analysis – although it covers the basic notions of functional analysis, it assumes the reader is somewhat acquainted with them. It is by no means a second course either: there are too many deep subjects that are not within scope here. Instead, having the basics under his belt, the author takes the time to carefully think through their fundamental consequences. In particular, the focus is on the notion of completeness and its implications, yet without venturing too far from areas where the description 'elementary' is still valid. The author also looks at some applications, perhaps just outside the core of functional analysis, that are not completely trivial. The aim is to show how functional analysis influences and is influenced by other branches of contemporary mathematics. This is what we mean by 'Functional Analysis Revisited.'
Introduction; 1. Complete metric spaces; 2. Banach's principle; 3. Picard's theorem; 4. Banach spaces; 5. Renewal equation in the McKendrick–von Foerster model; 6. Riemann integral for vector-valued functions; 7. The Stone–Weierstrass theorem; 8. Norms do differ; 9. Hilbert spaces; 10. Complete orthonormal sequences; 11. Heat equation; 12. Completeness of the space of operators; 13. Working in L(X); 14. The Banach–Steinhaus theorem and strong convergence; 15. We go deeper, deeper we go (into the structure of complete spaces); 16. Semigroups of operators; Appendix. Two consequences of the Hahn–Banach theorem; References; Index.
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