Principles of Mathematical Analysis
– by Walter Rudin
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Recommended: Principles of Mathematical Analysis
Author: Walter Rudin
Rudin, Principles of Mathematical Analysis: Infamously terse. Rudin likes to do things in the greatest generality and the proofs tend to be slick (i.e. rely on clever arguments that don’t really clarify the thing being proved). It’s thorough, it’s rigorous, and the exercises tend to be difficult. You won’t find any straightforward definition-pushing here. If you had a rigorous calculus course (like Courant’s book), you should be fine.
Kenneth Ross, Elementary Analysis: The Theory of Calculus: I’d put this book as a gap-filler. It doesn’t go into topology and is rather straightforward. If you learned the “cookbook” approach to calculus, you’ll probably benefit from this book. If your calculus class was rigorous, I’d skip it.
Serge Lang, Undergraduate Analysis: It’s a Serge Lang book. Contrary to the title, I don’t think I’d recommend it for undergraduates.
G.H. Hardy, A Course of Pure Mathematics: Classic text. Hardy was a first-rate mathematician and it shows. The downside is that the book is over 100 years old and there are a few relevant topics that came out in the intervening years.” – Posted by Epictetus at http://lesswrong.com/lw/3gu/the_best_textbooks_on_every_subject/ – by Books2Learn – 2018-02-27