Analysis II (Texts and Readings in Mathematics)

This book (as well as the 1st volume) comes out from the author's lecture notes. It's decent work but I would expect much more from a renowned mathematician.

The book takes a very formal approach. It starts with building the number systems, from natural numbers to real numbers. The proofs are generally more detailed than similar text books. The chapters on Lebesgue integration are good. The author explains why the Lebesgue measure exists, instead of just presenting the definitions and theorems, like most introductory books do. The remarks dotted everywhere are instructive, reflecting the author's broad knowledge in mathematics. No surprise, given Tao is one of the most versatile mathematicians in our time.

The bottom line is it's good but not exceptional. For introductory analysis book, I'll still choose Rudin's "Principles of Mathematical Analysis". It covers more topics yet it's more concise than these 2 volumes.

According to the back of the book, this book (and volume 1) are designed for an honors undergrad analysis class. This, however, is most definitely not the case. The book, especially volume 1, is quite basic: The first 150 pages are devoted to constructing the real numbers, and fundamentals (set theory, functions). The exercises aren't too bad, and the style of writing is nice. In volume 1 there are a few typos, and in Volume 2 there are a lot of typos (check Tao's UCLA page for an errata). My main complaint about the book is that Tao defines things his own way (which often differs from Rudin's book and others), so it is a little tough to supplement with other books.