A list of Rudin-style textbooks
I'm fond of baby Rudin: elegant presentation, "clever" proofs, certain terseness, difficult exercises, etc. I'm working up my enthusiasm to seek similar textbooks in other areas of math, e.g. linear algebra, abstract algebra, ODE, number theory. Below are some books I want to exclude:
Zorich's analysis—it certainly offers a higher-level view of analysis, but too wordy and explanatory for me.
Lang's Algebra—its selected proofs are concise and elegant, but it's intended to serve as a reference book.
Arnold's ODE—it's excellent, but not at all alike.
Please provide the closest one you've read. It would be appreciated if you can suggest books in branches that haven't already been touched on by other answers so as to diversify the list. I believe this list will be favorable to those who enjoy Rudin's style as much as I do. Please inform me if this is a duplicate, I'll close it immediately (I can't be sure even I've already looked it up). Thanks in advance.
Solution 1:
General Topology -- Both Engelking's General Topology (1989) and Borisovich's Introduction to Topology (1985) are very compact treatments of the subject and should be looked at by a challenge-seeking learner. Burisovich's text was originally written in Russian and published by Mir Publishers, and has the notorious Soviet rigour throughout.
Linear Algebra -- Lang's Linear Algebra is often overshadowed by his much more popular book "Algebra", but it is an amazingly written text on linear algebra of arbitrary vector spaces. I learned linear algebra from the 2nd edition, and it was unbelievably well written and rigorous. It is not as difficult as his book on abstract algebra, but it is by no means easy. (Note: The later editions changed the writing style up a bit so I would recommend looking at one of the first few editions). Also, Greub's Linear Algebra (the predecessor of his Multilinear Algebra book discussed below) is an extremely comprehensive, and tersely written, treatment, which is accompanied by difficult exercises
Mathematical Analysis -- Dieudonné's Treatise on Analysis (translated from the French Éléments d'analyse) is a 9 volume rigorous treatment of the subject which has the same slick writing style as Rudin's books. It should be noted that Dieudonné was a founding member of the Bourbaki group.
Multilinear Algebra -- Northcott's Multilinear Algebra is very well written and entirely rigorous, it functions as a very good supplement to Bourbaki's Algebra I (Chapters 2 & 3 discuss similar content). Obviously, Bourbaki is extremely rigorous, and I would say far surpasses Rudin's writing style in terms of sophistication. (I am also fond of Greub's Multilinear Algebra, but its scope is restricted mostly to vector spaces, and does not discuss modules over rings)
Solution 2:
Functions of One Complex Variable, by J. B. Conway. I found the organisation of the material and the proofs of the results very reminiscent of Rudin's. The exercises may not match up to the ones in Rudin though.
Algebra: A Graduate Course, by Martin Isaacs. This is a much better candidate for an algebra book that is Rudin-like than Herstein's classic text, in my opinion. The proofs in this textbook are truly slick and beautiful; this is in stark contrast to the combinatorial proofs in Herstein.