A first course in functional analysis. (English) Zbl 1367.46001

Shalit, Orr Moshe

Boca Raton, FL: CRC Press (ISBN 978-1-4987-7161-0/hbk; 978-1-4987-7162-7/ebook). xv, 240 p. (2017).

This book aims to introduce basic notions and methods in functional analysis (with an emphasis on some important aspects of Hilbert spaces and their operators) for undergraduate students in mathematics. The key point is that the author nicely presents an introduction to functional analysis with a slight application of \(L^p\) spaces (without measure theory). More precisely, he first introduces the space \(L^2[a, b]\) as the completion of the space of piecewise continuous functions on \([a, b]\) equipped with the norm \(\|f\|_2=\left(\int_a^b|f(t)|^2\, dt\right)^{1/2}\) and then uses it to present some results on Fourier series and the mean ergodic theorem.
The book includes 13 chapters: Introduction and the Stone-Weierstrass theorem; Hilbert spaces; Orthogonality, projections, and bases; Fourier series; Bounded linear operators on Hilbert space; Hilbert function spaces; Banach spaces; The algebra of bounded operators on a Banach space; Compact operators; Compact operators on Hilbert space; Applications of the theory of compact operators; The Fourier transform; The Hahn-Banach theorems.
The author studies Hilbert spaces before Banach spaces and so has to state some notions and theorems first for Hilbert spaces and then again for Banach spaces. For example, the notion of bounded linear operators is given both in Definition 5.1.1 and Definition 7.2.1. Of course, as he argues in the preface, besides other reasons, he does not think that there is a point in introducing greater generality before one can prove significant results in that generality. He also proves the Stone-Weierstrass theorem in the first chapter (introduction) to show the inadequacy of countable Hamel bases in functional analysis, but the reviewer thinks a teacher may postpone the proof of this theorem to, e.g., Section 8.1, where the algebra of bounded operators is introduced.
Several extensive exercises in the chapters clarify the text and help the readers to extend their knowledge of functional analysis. In addition, an appendix containing some significant material from metric and topological spaces used in the book is given.

Reviewer: Mohammad Sal Moslehian (Mashhad)

MSC:

46-01	Introductory exposition (textbooks, tutorial papers, etc.) pertaining to functional analysis
47-01	Introductory exposition (textbooks, tutorial papers, etc.) pertaining to operator theory

Keywords:

Stone-Weierstrass theorem; Hilbert space; orthogonality; projection; basis; Fourier series; Hilbert function space; Banach spaces; bounded linear operator; compact operator; spectrum; spectral theorem; Fourier transform; Hahn-Banach theorem