This book is a tour de force, proceeding in a mere 110 pages from essentially no knowledge past calculus to a proof of a famous theorem of Wiener on absolutely convergent Fourier series. The present volume is a 2013 Dover reprint of the 1966 Gordon and Breach work.
Everything is explained adequately and the exposition is not rushed or difficult to follow. It reaches its goal by staying totally focused and not taking any detours. (Well, hardly any detours. It does have coverage of integral equations, in particular the Fredholm and Volterra forms, but this only takes two or three pages.) The book is weak on motivation, and it takes a moderate amount of faith to keep reading. It follows the Theorem–Proof model of mathematical exposition, with few examples or digressions, although there are exercises at the end of each chapter.
The typesetting is a little shaky, with some irregular spacings and an idiosyncratic use of the set membership symbol for the small quantity epsilon. Most students wouldn’t notice these things. There are also a few incomplete sentences and wrong words. It seems the book was not proofread carefully, but since it happened fifty years ago, we are stuck with it.
I like this book, but I am uneasy about its narrow focus. I think for a first course I would be happier with something like Saxe’s Beginning Functional Analysis, that shares the streamlined approach of Davis’s book but has more breadth.
Allen Stenger is a math hobbyist and retired software developer. He is webmaster and newsletter editor for the MAA Southwestern Section and is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis. He volunteers in his spare time at MathNerds.org, a math help site that fosters inquiry learning.
|1. Set Theoretical Preliminaries|
|2. Normed Linear Spaces and Algebras|
|3. Functions on Banach Spaces|
|4. Homomorphisms on Normed Linear Spaces|
|5. Analytic Functions into a Banach Space|