Looking for a book that picks up where Understanding Analysis by Abbott left off?

Question

I'm currently going through the Understanding Analysis text by Abbott and was interested in what typically comes after once I finish going through this book. Would multivariable analysis of some sort come next? If so, what book would you recommend? Or is there more "single-variable" analysis left to be done?

Edit: I noticed the downvote, would appreciate any advice on how I can improve my answer.

Answer 1

Another good choice is Ethan D. Bloch " The real Numbers and Real Analysis" Springer

Answer 2

Since Abbott's book is rigorous, I think you are ready to start more advanced topics like metric spaces.

The classic book to read then is the book "Principles of mathematical analysis" by Walter Rudin (also called Baby Rudin). But I believe this text is not meant to study from, but rather to master the material from.

Therefore, I would first read the book "Mathematical analysis" by Tom Apostol. This treats more or less the same material as Rudin, but in a much more gentle way. If you mastered this book, you can go read Rudin.

Answer 3

I recommend looking at Pugh's Real Mathematical Analysis because, among other features, it's excellent expository merits are similar to those that Abbott's book has. Although a nontrivial amount of the material in Pugh's book will be familiar to you, Pugh's treatment is at a slightly higher mathematical maturity level and the scope of the applications covered (in the text, and especially in the exercises) is far beyond that of Abbott and will expose you to many new topics. Also, as you can see from the table of contents, you'll get a good exposure to multivariable topics.

Answer 4

I happen to really like large sections of Koerner's A Companion to Analysis: A Second First and First Second Course in Analysis. It starts off with analysis in $\mathbb R$ then moves to $\mathbb R^n$ and eventually metric spaces. There are even a few bonus problems on complex analysis near the back. Koerner's book's corrections (and many solutions) are available on his website https://www.dpmms.cam.ac.uk/~twk/ . The goal of the book in some sense is to give you a new perspective on real analysis after you've already taken a course on it.

That said, you ask whether there is more "'single variable' analysis left to be done" -- an alternative and natural path at this point would be a book on complex analysis... there are many recommendations on this site.