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Real Analysis

Fon-Che Liu
Oxford University Press
Publication Date: 
Number of Pages: 
Oxford Graduate Texts in Mathematics 26
[Reviewed by
Michael Berg
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This is a rather unusual book on real analysis, or maybe it isn’t all that unusual but I’m just getting old. I grew up, many, many moons ago, with the likes of Royden and Rudin, and maybe a few other contenders for a first year graduate real analysis text — in fact, being a goofy and self-willed youth, I often experimented with other sources (generally with less than desirable results: if it ain’t broke, don’t fix it) — but the two mainstays just mentioned were ubiquitous. And it seemed like this state of affairs was chiseled in stone. But it’s not so, of course, and I doubt whether Royden and Rudin are still as popular today as they once were. Just based on spot checks and chats with colleagues, it seems like a generational thing: folks within, say, a decade of my age, are very familiar with these books. I’m not so sure about the younger people.

Well, all right, but it is still the case that the arrangement and sequence of topics in such a course are pretty much invariant, isn’t it? Measure theory, the Lebesque integral, functional analysis … plus ça change, plus c’est la même chose, non? I guess it’s indeed no(n). At least if the book under review is representative. Liu starts with a chapter on “Introduction and Preliminaries” which includes such things as double series, metric spaces and normed linear spaces, \(l^2(\mathbb{Z})\) (wow), compactness, connectedness, and local compactness. OK, why? Well, because the author wants to “prepare the reader for a more systematic development in later chapters of real analysis through some introductory accounts of a few specific topics … [and] to acquaint [certain] readers with the fundamentals of abstract analysis.” Aha! Now it’s beginning to make sense, and indeed in short order c’est la même chose after all: after “A Glimpse of Measure and Integration” Liu does “Construction of Measures” and “Functions of Real Variables.” The latter chapter includes a discussion of Riemann as well as Lebesque integrals, and, yes, Riemann-Stieltjes, too. There’s the important business of product measures and Fubini, as well as some coverage of nuts-and-bolts stuff like changing variables in multiple integrals, done at a very sophisticated level: we’re doing Lebesgue integration with some gusto here. It’s much more than what one sees in third semester calculus (which material is of course present incognito). Cool.

Now that we are properly airborne we get mainstay material, of course: there are certain non-negotiables in graduate real analysis. In the last three chapters Liu gives us, for example, Baire category, open mapping, closed graph, Hahn-Banach, Lebesgue-Nikodym (which, I guess, is Radon-Nikodym: according to Wikipedia we get the Lebesgue-Radon-Nikodym Theorem. News to me…), and then a separate chapter on \(L^p\)-spaces. The last chapter is devoted to Fourier integrals and Sobolev spaces. So, yes, it’s quite a menu, and certainly compares well with other real analysis books, even the older sources mentioned above.

Liu taught all this material in a first year graduate analysis course at National Taiwan University for over 30 years, so it’s time-tested by someone who’s done his time in the pedagogical trenches. By the way, I note from the Mathematics Genealogy Project that Liu’s advisor at Purdue was Casper Goffman, who wrote my single favorite undergraduate analysis book. So it’s all good, as they say nowadays, and so is the book under review. A good pace, many (very many) exercises, and a very nice presentation. Kudos to Prof. Liu.

Michael Berg is Professor of Mathematics at Loyola Marymount University at Los Angeles, CA.

1. Introduction and Preliminaries
2. A Glimpse of Measure and Integration
3. Construction of Measures
4. Functions of Real Variables
5. Basic Principles of Linear Analysis
6. Lp Spaces
7. Fourier Integral and Sobolev Space Hs
8. Postscript