The reference requests for analysis books have become so numerous as to blot out any usefulness they could conceivably have had. So here come another one.
Recently I've began to learn real analysis via Rudin. I would do all the exercises, and if I was unable to do them within a time limit (usually about 30 min) I would look the answers up. Combined with the excellent online lectures by Francis Su, I made rapid progress. Encouraged I now intend to self-study analysis II and function theory. However apart from its uninformative and dry style, Rudin's does not cover everything I intend to study.
After searching for a suitable textbook, I was particularly attracted to Analysis I&II by Terry Tao. His breadth of knowledge and his nack for clear exposition are famous but I particularly like that he starts from the very beginning and builds it up from there, as well as putting real analysis inside a greater unified whole. His books would cover exactly what I intend to study. For instance, he covers fourier series, which Rudin's doesn't.
However after searching for hours I've been unable to find any solutions sets. (apart from a few on the earliest chapters). It is my experience that is almost impossible to self-study a subject thoroughly without solutions or constant feedback, even with an outstanding textbook. Which leaves me with few options:
- Proceed with Rudin's, perhaps with some supplementary book.
- Try to work with Terry Tao's Analysis I&II without solutions.
- Find a different book altogether that is both comprehensive and readable as well as having at least a partial solution set.
I know a lot of people will recommend Rudin but I have to doubt their experience with self-study: yes it is possible to learn directly from Rudin but it's painful and slow. And quite frankly I feel that a lot of people have poured a lot of time and effort in Rudin and feel that more than teach them analysis it has brought them mathematical maturity. That is all well and good but it's not what I'm interested in.
Another idea would be to get both and read Tao, while doing the exercises in Rudin's. I don't think that would be a good idea however, a lot of theorems in Tao are left to the reader and the pace and coverage of both books are very different. In general I dislike getting more than one book.
Does anyone know of an extended (partial) solutions set to Terry's analysis I&II or otherwise a reference for another book that would be suitable?