The reference requests for analysis books have become so numerous as to blot out any usefulness they could conceivably have had. So here comes 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?
I learned most of my analysis from Tao's excellent volumes. The OP is correct that Tao starts at the beginning with defining the number systems. So I believe chapter 5 or Tao is equal to chapter 1 of Rudin, or something like that.
I did not have too much trouble doing the proofs and exercises for the initial chapters of Tao because he really develops the ideas slowly. So that was good. It was only when he started to address the point set topology stuff that I started to have a harder time. So I actually decided that I would do a little more exploration of Point-Set topology on my own. So I read Munkres' book and then did the exercises in the Schaum General Topology outline. That was a good combination.
After I had a better handle on rigorous topology, then I really had not trouble following the rest of Tao's analysis. I think I ready through Tao's analysis and use the Schaum outline in Advanced Calculus, and in Real Variables for doing problems. Once I was able to do the problems in Schaum, then I had much less trouble doing the problems in Tao. In many cases the problems were similar.
But that is just how I approached it. I hope this info helps others pursuing Analysis. It really is a beautiful subject and I fear that people often get psyched out by claims of its difficulty.