Lectures and Exercises on Functional Analysis (Translations of Mathematical Monographs, 233)

This book contains a unique exposition intended to serve as an introduction to functional analysis. Topics covered include normed spaces and bounded operators, Banach spaces, polynormed spaces and distributions, compact operators, $C^\*$ algebra, spectral theorems, Fourier transform, and more. A distinguishing feature of the book is the wide use of the language and elementary constructions of category theory, which are explained in the opening chapter of the book. Among nonstandard topics discussed in the book are the theory of Banach tensor products, basics of quantum functional analysis, and Borel operator calculus. General definitions and main results are supplemented with many examples and exercises. Prerequisites for the main part of the book include standard undergraduate courses in algebra and analysis. It is suitable for graduate students and researchers interested in functional analysis. Functional analysis is the branch of mathematics concerned with the study of spaces of functions. Helemskii (Moscow State U., Russia) presents this text as an introduction to the field, devoting the bulk of it to the classical functional analysis of normed, Banach, and Hilbert spaces, but also covering polynormed spaces and some of their applications, as well as those elements of the theory of Banach algebras important to the study of spectra. The treatment differs from others on the same topic in that it emphasizes the categorical nature of the fundamental construction and results. The text assumes knowledge of linear algebra, the foundations of real analysis, and the elements of the theory of metric spaces found in advanced analysis courses. Annotation 2006 Book News, Inc., Portland, OR (booknews.com) The book is based on courses taught by the author at Moscow State University. Compared to many other books on the subject, it is unique in that the exposition is based on extensive use of the language and elementary constructions of category theory. Among topics featured in the book are the theory of Banach and Hilbert tensor products, the theory of distributions and weak topologies, and Borel operator calculus. The book contains many examples illustrating the general theory presented, as well as multiple exercises that help the reader to learn the subject. It can be used as a textbook on selected topics of functional analysis and operator theory. Prerequisites include linear algebra, elements of real analysis, and elements of the theory of metric spaces.