As the first edition of this classic textbook was published in 1963, it is worth a look to see what changed in forty-seven years (and now a posthumous co-author), and why. First, not surprisingly, the book became much larger: it is now 505 pages long and has sufficient material for three semesters. The original was 284 pages, and was probably meant for two semesters.
This would suggest that most parts of the book got extended. This is largely true, but there is one notable exception to that rule. Set Theory was an independent first chapter of the first edition, which was then followed by a chapter on the real number system. In the latest edition, set theoretical preliminaries are only discussed in a very brief introduction. Perhaps the reason for this is that comtemporary first-year graduate students (the target audience of this book) are more familiar with concepts such as Zorn’s lemma or the Axiom of Choice than their peers of the 1960s were.
Topics whose coverage wastly expanded since the first edition are Lp spaces, Banach Spaces, and Linear Operators. The number of exercises increased very significantly: according to the preface, it grew fifty percent since the third edition. There are still no exercises with solutions.
Despite all these changes, the tone of the book is extremely similar to that of previous editions. Each section is introduced by a short two or three paragraph introduction outlining what will be covered. The rest of the section closely follows that plan, often to the point of being terse and not having enough examples. (Perhaps this is the price to pay if one is to keep a three-semester book to 500 pages.)
On the whole, the reviewer believes that, for good or ill and 47 years notwithstanding, this is still essentially the same book as it was in 1963.
Miklós Bóna is Professor of Mathematics at the University of Florida.