Lectures and Exercises on Functional Analysis
This book on functional analysis is written with a considerable use of the language of category theory. The introductory chapter is devoted to foundations (metric and topological spaces, categories, morphisms and functors). The next chapters cover basic elements of functional analysis: normed spaces and operators (and the Hahn-Banach theorem and an introduction to quantum functional analysis) and Banach spaces (including tensor products). A special chapter presents the theory of polynormed spaces (basically locally convex spaces), weak topologies and the theory of distributions. Further chapters are devoted to compact and Fredholm operators and spectral theory (among others C*-algebras and Borel functional calculus). The final chapters deal with Fourier transforms and fundamental harmonic analysis on groups. The book contains many examples and exercises of different levels of difficulty. It requires only a basic knowledge of linear algebra, elements of real analysis, and metric spaces. It can be recommended to a broad spectrum of readers, to graduate students as well as young researchers.