Conway J.B. A Course in Functional Analysis

Springer-Verlag, 1985. — 404 pages.This book is an introductory text in functional analysis, aimed at the graduate student with a firm background in integration and measure theory. Unlike many modern treatments, this book begins with the particular and works its way to the more general, helping the student to develop an intuitive feel for the subject. For example, the author introduces the concept of a Banach space only after having introduced Hilbert spaces, and discussing their properties. The student will also appreciate the large number of examples and exercises which have been included.Review
The author begins his Preface with the statement, "Functional analysis has become a sufficiently large area of mathematics that it is possible to find two research mathematicians, both of whom call themselves functional analysts, who have great difficulty understanding the work of the other. " As a corollary, it is possible to find two books or courses that bear the title "Functional Analysis" and have little in common. In the present book the author has decided to follow what he calls the "common thread" of functional analysis, namely the existence of a linear space with a topology. Throughout, the emphasis is placed on the study of bounded operators on a Hilbert space. The author begins by discussing the basic properties of Hilbert spaces and operators on them. He follows this with Banach spaces and locally convex spaces. Chapters are devoted to the weak topology on a Banach space and the weak star topology on its dual, bounded linear operators on a Banach space, Banach algebras, C* algebras, normal operators on a Hilbert space (bounded and unbounded) and Fredholm theory for bounded operators on a Hilbert space. Many examples are given, including some from other branches of mathematics. Numerous exercises are provided. The author has also included several important applications and topics such as Fourier transforms, invariant subspaces, Sturm-Liouville systems, Banach limits, Runge's theorem, ordered vector spaces, inductive limits, distributions, the Stone-Cech compactification, the Krein-Milman theorem, the Stone-Weierstrass theorem, the Schauder fixed point theorem, the Ryll-Nardzewski fixed point theorem, Haar measure, the Krein-Smulian theorem, and moments.