Functional analysis as a discipline is about a hundred years old. Its beginnings go back at least as far as the work of Fréchet, Hilbert and Banach. The original questions that motivated it arose largely from the study of differential and integral equations. Over the years the subject has developed as a curious mixture of hard and soft analysis, with the book under review definitely tipped in the direction of the former.
As the subject of a course in graduate school, the contents of functional analysis are fuzzily defined, and there is likely considerable variation from school to school and year to year. Some of the material commonly considered fundamental functional analysis (Hilbert and Banach spaces, as well as Hahn-Banach, Banach-Steinhaus, Open Mapping, and Closed Graph theorems) is now routinely included in first year graduate real analysis courses. If there is indeed a core, it would probably be the study of operators on Hilbert space and the spectral theorems. Curiously, functional analysis is largely absent in The Princeton Companion to Mathematics; there is no mention of it among the “Branches of Mathematics,” nor does it appear in the index.
The book under review, known in my graduate school days simply as Riesz-Nagy, was originally published in 1955 and has been available in a Dover edition since 1990. For a while — including the period I was in graduate school — it was used as a textbook, but it is probably used more widely as a reference. There are no exercises. Even though some of the terminology is a little old-fashioned, the exposition is wonderfully clear. It is not possible to read this book without feeling that one is in the presence of masters.
There are two parts of the book. The first, “Modern Theories of Differentiation and Integration,” includes much that is now included in graduate real analysis courses. It begins with Lebesgue’s theorem on the derivative of monotonic functions, pursues the consequences of that result, and then moves on to the Lebesgue integral, Lp spaces (devoting special attention to L2) and linear functionals. All work is done explicitly in Euclidean space and the treatment of measure theory as such is very limited.
The second part, “Integral Equations, Linear Transformations,” is the meat of the book and accounts for about two-thirds of its length. The authors say, “The general theory of integral equations is the work of our century.” They introduce integral equations via the examples of Fourier, Abel and Volterra. Next, they discuss solution by successive approximation and the Fredholm alternative. An introductory chapter on Hilbert and Banach spaces precedes four chapters on the spectral theory of bounded and unbounded operators on Hilbert space.
This book is a marvelous reference and a model of clarity. It deserves a place on every analyst’s bookshelf.
Bill Satzer (firstname.lastname@example.org) is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.