Functional Analysis Revisited
An Essay on Completeness

A unique treatment of functional analysis that shows its interactions with other branches of contemporary mathematics.

Adam Bobrowski (Author)

9781009430913, Cambridge University Press

Hardback, published 4 July 2024

250 pages
22.9 x 15.2 x 1.6 cm, 0.532 kg

'This non-standard book can serve as a supplement to an introductory functional analysis course. Centred around the fundamental notion of completeness, it introduces a variety of basic notions and tools, and covers a solid range of material presented in a lively and vivid way. The emphasis has been placed on applications of abstract concepts, including the rudiments of operator semigroups, partial differential equations, and probability theory. The exposition is friendly and fresh, and many insightful comments facilitate digesting the material. Each chapter is accompanied by helpful summaries and carefully selected exercises. The book would be useful for beginners aiming to improve their understanding of the subject, for lecturers willing to refresh their routine courses, and it would be of value to anyone who likes elegant expositions of important mathematical theories.' Yuri Tomilov, Institute of Mathematics, Polish Academy of Sciences

'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.

Subject Areas: Calculus & mathematical analysis [PBK]