[tashi]

Well, if you're going to take the high road and give him the benefit of the doubt, I won't argue further. Maybe I'm just being cynical.

But in the review he wasn't just talking about "people who make maths the primary focus of their study." If he had just been talking about students' areas of interest or work ethic I would never have objected. It was specifically this: "One should pick one's audience carefully... and treat these gifted kids like apprentices." In my experience that approach misses a lot of talented people who were different enough not to get matched by the "gifted" filter.

[aoki]

most serious US math programs have honors sequences that are aimed at students who intend to go for a PhD and who are often starting with post-calc-I/II/III courses from the very beginning. tao says right in the introduction that this text is for such an honors sequence at UCLA. so, yeah, this book is explicitly written for undergrads who are being groomed to enter elite graduate programs because they did the work (before college) to fit that profile.

in the US, i believe the real analysis course for non-honors courses at R1s is often based on something like ross, abbott, or bartle & sherbert, whereas for non-R1s (where most math majors will be teachers) it may be based on something like lay or wade instead. these books are more accessible to students with less mathematical maturity.

i think what the reviewer is saying is that it would be a mistake to use this book in one of those non-honors courses with a poor faculty-student ratio. even if you do have some students who have the interest and ambition, you'd be doing a disservice to the rest. you'd be exceeding the level of interest for most and unable to support anybody adequately.