Books like Stephen Abbott's Understanding Analysis

Solution 1:

I know it has been two years past this question, but recently I took a course on Analysis and read Understanding Analysis. I loved the book and started looking for other similar ones, and that’s how I got to your question.

Up until now, I found just one other book that I thought was similar to Abbott’s approach. The book is The Cauchy-Schwarz Master Class - An Introduction to the Art of Mathematical Inequalities, by Michael Steele.

In this book, the author shows the different relations and applications of a series of inequalities, while giving you an intuition behind them. All this starting from Cauchy-Schwarz inequality. It’s pretty good.

If I find any new book that is similar, I’ll update the answer.

As promised, I found another book. Measures Integrals & Martingales, by Ren L Schilling. This one is about measure theory and is just great. The author also provides a free solution manual to all question, hence it is just perfect for self-study. Also, the explanations are very clear and perfect for undergraduates or first year graduates.

Solution 2:

This is a late answer ($3$ years late!), but I thought I'd put my two cents in just in case somebody ever stumbles across this question.

One book that matches your description perfectly (and, I'd argue, is even more successful at what it does than Abbott's) is Michael Sipser's Introduction to the Theory of Computation. Instead of assuming the definition-theorem-proof structure standard in mathematical literature, Sipser splits his proofs into two parts: a "Proof Idea", which explains the motivation behind each proof and fleshes out the details, followed by the condensed Proof to provide the rigorous formalism.

Exercises are similarly segmented: each chapter is followed by an "Exercises" section, providing the student with an opportunity to practice basic computations and apply the learned definitions to some elementary results, and a "Problems" section, which compiles more challenging results. As is standard, some important theorems that the main text couldn't accommodate are relegated to the Problems section. The book is perfectly linear and the level of understanding demanded throughout is very even.

Perhaps the only piece of critism I can think of for this book is that it isn't advanced enough. But (in case of persisting interest) this is easily rectified by following it up with a more advanced text.