Real Analysis

Below is a list of reviews of books on real analysis.

Principles of Mathematical Analysis
Rudin, W. (3rd Ed. 1976). ISBN 9780070856134.
Prerequisites: Proof writing and single variable calculus.

Rudin’s book is infamous and widely used, but it is also terse and difficult. Rudin presents material in reasonable generality (through metric spaces) and spends less time on motivation in favor of definition-theorem-proof style writing. Rudin’s proofs are often very clean, but sometimes fail to convey how you would have come up with the proof yourself. The exercises are great, they scale well in difficulty and sometimes guide you to proving relevant results. The best parts of Rudin are Chapters 1-8, including the reals, metric spaces, sequences and series, continuity, differentiation, Riemann integration, function convergence, and special functions. The remaining chapters on differential forms and Lebesgue integration are poorly written - use a different book after Chapter 8 of Rudin.

Real Mathematical Analysis
Pugh, C. C. (2nd Ed. 2015). ISBN 9783319177700
Prerequisites: Proof writing and single variable calculus

Pugh’s book is the antithesis to Rudin’s. It covers material in good generality (metric spaces and basic ideas of general topology), but unlike Rudin, it is written with tons of motivation, pictures, and descriptions of both the objects and the proofs. The downside of this is that Pugh’s proofs can feel less rigorous and clear. There are even a few “proofs by picture.” Pugh also covers a few esoteric extra topics on top of the ones covered by Rudin, e.g. theory of Cantor sets. Pugh has excellent exercises ranging in difficulty, including some “3-star” problems that he himself can’t solve. Pugh’s sections on differential forms and Lebesgue integration are better than Rudin’s, but it is still best to read a book dedicated to these and to stop reading Pugh after Chapter 4.

Analysis I + II
Amann, H., Escher, J. (2005, 2008). ISBN I 9783764371531, ISBN II 9783764374723.
No formal prerequisites

This is the standard book used for teaching mathematical analysis in Germany and could be somewhat described as the German Rudin. It is available fully translated in English, which already speaks for its quality. The first chapter (of the first book) starts with an introduction to proofs and somewhat unorthodoxly with some group, ring and field theory as well as some linear algebra. This chapter can be skipped and used as a reference when needed for the later things. Thorough treatment of all topics covered and very nicely written with a ton of content (the second book also includes a treatment of complex analysis). Includes a ton of good exercises for all levels.

An Introduction to Analysis
Wade, W. (4th ed. 2009). ISBN 9780132296380.

I don’t see this book mentioned often, but it’s one of my analysis professor’s favorites and one of mine too. It covers the contents of a standard sequence in single-variable, then multivariable analysis. The definitions are well-motivated, and the textbook is much friendlier to read than a text like Rudin but covers more material and moves faster than a text like Abbott. The formatting of the text is very clean, and I also found that it did a good job of emphasizing important points. The progression of topics is very natural, and it also contains some optional topics like Fourier series that often aren’t included in a first course in analysis. Perfect for a student learning analysis for the first time.