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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.

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    $\begingroup$ I love all books of Rudin. If you have enjoyed Baby Rudin then at some stage you should read all of his books. $\endgroup$ – Kavi Rama Murthy Dec 20 '18 at 7:45
  • $\begingroup$ I will. But it only covers topics in analysis, unfortunately. $\endgroup$ – YuiTo Cheng Dec 20 '18 at 7:46
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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)

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  • $\begingroup$ About Borisovich's book, again, it's out of print... $\endgroup$ – YuiTo Cheng Dec 21 '18 at 3:15
  • $\begingroup$ I purchased a used copy online a few years ago for about $20 (USD), but a pdf can be found easily as well. It is currently for sale on amazon and abebooks, etc, under Borisovich - Introduction to Topology (1985) - Mir Publishers $\endgroup$ – Victoria M Dec 21 '18 at 3:18
  • $\begingroup$ Yes it is Borisovich, I fixed the typo right before you commented. It has been corrected in the original answer. $\endgroup$ – Victoria M Dec 21 '18 at 3:20
  • $\begingroup$ Do you think the third edition of Lang's Linear Algebra is actually worse than the second one? I myself find it pretty easy so it doesn't fit my requirement here. $\endgroup$ – YuiTo Cheng Dec 21 '18 at 5:23
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  • 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.

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Functional Analysis by Michael Reed and Barry Simon.

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    $\begingroup$ Rudin also wrote a book on functional analysis...So could you explain how this book differs in content from grandpa Rudin? $\endgroup$ – YuiTo Cheng Dec 21 '18 at 3:24
  • $\begingroup$ @YuiToCheng the book is written in a terse style similar to Rudin and has really good and challenging exercises. It also has an excellent treatment of spectral theory. $\endgroup$ – Mustafa Said Dec 30 '18 at 9:03
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For Linear Algebra, I suggest Katsumi Nomizu's Fundamentals of Linear Algebra.

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    $\begingroup$ This is interesting. I didn't know Nomizu wrote a linear algebra book. However, even Google books doesn't have a preview of the book. $\endgroup$ – Jacky Chong Dec 20 '18 at 8:57
  • $\begingroup$ As it stands, this answer is just a comment. May I suggest that you expand this answer to include more details? $\endgroup$ – Brahadeesh Dec 20 '18 at 9:13
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    $\begingroup$ @Brahadeesh I disagree. It is implicit in my answer that Nomizu's textbook is a Rudin-style textbook. That makes what I wrote a complete answer. $\endgroup$ – José Carlos Santos Dec 20 '18 at 9:19
  • $\begingroup$ It is definitely implicit, but I stand by what I said earlier. $\endgroup$ – Brahadeesh Dec 20 '18 at 9:23
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    $\begingroup$ So perhaps in addition to this book, can you suggest an in-print one? $\endgroup$ – YuiTo Cheng Dec 21 '18 at 0:53
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For algebra,try Herstein's Topics In Algebra.

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    $\begingroup$ As it stands, this answer is just a comment. May I suggest that you expand your answer to include more details, such as why you feel that this book has the characteristics that the OP feels Rudin's books possess? $\endgroup$ – Brahadeesh Dec 20 '18 at 9:11
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    $\begingroup$ I've read it and it's beautifully written! But besides the degree of sophistication of the exercises, I'm unsure whether it's Rudin-style... $\endgroup$ – YuiTo Cheng Dec 20 '18 at 9:36
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Your description instantly reminded me of Introduction to Commutative Algebra by Atiyah and MacDonald.

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