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Real Analysis

Halsey Royden
Prentice Hall
Publication Date: 
Number of Pages: 
[Reviewed by
Fernando Q. Gouvêa
, on
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akirak's picture

Royden's is still one of the most widely used texts on the theory of measure and integration in print. As such, it's a perfect example of a fascinating phenomenon in academia: the rubber-stamping of texts for courses that were written by Ivy League Gods. Graduate complex analysis? "Get Alfhors." Topology? Munkres. And if any struggling student dares question it, the professor makes them feel like they have an IQ of 6. If the student can't handle the professor's favorite text from his student years at MIT, he is too dumb to learn analysis.

There are many factors leading to this elitist rubber stamping, not the least of which is that professors at research based power schools or with that kind of training usually think teaching is far from their first priority. But there's something else at work here, a kind of academic canonization of textbooks based on the research pedigree of the authors. Alfhors, for example,was a giant of function theory at Harvard. Sadly, anyone in academia can tell you a great creator does not a great teacher or expositor make — often just the opposite, in fact. Simply put,a lot of these guys have no idea how they make the discoveries that they do. This affects their textbook writing — the books are written far more to impress their colleagues than to educate students. Of course, there are exceptions — the late great Serge Lang at Yale and John Milnor of SUNY Stony Brook both earned their reputations as both great teachers and scholars. But they are the exceptions.

Which finally brings me to Royden's Real Analysis. With the possible exception of Rudin's Real and Complex Analysis, this may be the single most assigned text for the all-important graduate real analysis course. And frankly, I have no idea how anyone can learn real analysis from this thing. The one thing that strikes you most about it is that Royden seems to shunt the most difficult results to the execises. In each section, he gives a couple of careful definitions, a few theorems and lemmas, and then the problem sets are practically small research assignments! The most egregious example I found is this: He proves the additivity of measures in pedantic detail, but leaves the Tychonoff Theorem on product spaces as an exercise.

There are also some strange comments in the book that don't quite make sense:For example, he defines a function in the the introductory chapter as a rule between sets; in a footnote, he gives the Cartesian product definition of function and criticizes it as having problems in certain cases, such as the identity function, since it is not a true subset of a Cartesian product. Say what?

Scholars are sadly no more enlightened then the rest of human society — which is probably why after four decades and the death of its author, this book keeps selling, even at such an outrageous price. It is used as a test and a filter for graduate students. This is sad, since there are so many superior texts on this beautiful and critical subject — Angus Taylor's masterwork, General Theory of Funcitons and Integration (in Dover paperback,no less!), Hienz Baur's very modern and probability geared Measure and Integration, the old fashioned but wonderfully clear presentation in R. Wheeden and A. Zygmund's classic Measure and Integral, for example. More recently, Stein and Shakarchi have produced a beautiful — and much less expensive — Real Analysis text in the Princeton Lectures in Analysis series.

[Andrew Locascio is currently a graduate student at Queens College Of The City University Of New York.]

akirak's picture

I used Royden for an advanced undergraduate course in Lebesgue integration, and found it quite useful. Yes, it belongs to a tradition of concise and no-nonsense texts that can be quite intimidating to the student. On the other hand, when they are done right, such texts reward the effort that needs to be put into reading them.

However, I agree completely with the two main points of the review above: one shouldn't adopt a text simply because one used it in graduate school or because it is well-known, and the prices on many of these classic texts have become ridiculously high. In the case of measure theory and Lp spaces, I agree that Stein and Shakarchi have provided something special in their Princeton Lectures series and I too would encourage people to look around to see if there isn't a text that is better - both mathematically and pedagogically - than the old standard.

(Outis Niemand, Dystopia College)