As real analysis books go, this one is very abstract. Much of it deals with topics that would usually be considered functional analysis, such as the Hahn-Banach Theorem and the Open Mapping Theorem, along with quite a lot on dual spaces. The most concrete topic in it is the Fourier transform, and even this is done on locally-compact abelian groups rather than on the real line.
The book is intended as a second graduate course in analysis, after a course on measure and integration. It originated as a blog that posts lecture notes for a course 245C in real analysis at UCLA that uses Folland’s Real Analysis: Modern Techniques and Their Applications (Wiley, 2nd edition, 1999) as a secondary text. The blog entries have been cleaned up somewhat here and the result does indeed read like a textbook, not like lecture notes or a blog. This is the first of two volumes from the third year of Tao’s blog; the second volume will cover a variety of subjects. The first two years have already been published in three volumes: Structure and Randomness: Pages from Year One of a Mathematical Blog, Poincaré’s Legacies, Part I: Pages from Year Two of a Mathematical Blog, and Poincaré’s Legacies, Part II: Pages from Year Two of a Mathematical Blog.
Overall I was disappointed in this book. It is a competent treatment and does include a number of interesting things, but it is also ordinary. It doesn’t have the kind of brilliancies and insights that you would expect from an mathematician of this author’s caliber. The first half of the book is a fairly conventional course in functional analysis. Things improve somewhat in the next quarter of the book, which turns these very general theorems to more particular structures, in particular a long chapter on Fourier transforms and a shorter one on distributions. The last quarter of the book is a potpourri of sidelines from the main text and you may or may not like it based on your interests; it has quite a lot on the Axiom of Choice, and several short articles on specialized topics.
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.