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I'm attempting to self-study real analysis and was looking at the popular textbook recommendations. So far, I've seen Abbott and Cummings for beginners. With Rudin, Stein, Bartle, and Zorich for more advanced books.

My question is : What book is the sort of "Bible of Real Analysis" or reference book, where if one knows the book from cover to cover he would have a colossal grasp on the subject.

To clarify my question, I mean a textbook that would cover the whole surface of analysis after learning it from a more elementary book such as Abbott or Zorich.

fiftytwo
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  • https://math.stackexchange.com/questions/62212/good-book-for-self-study-of-a-first-course-in-real-analysis – BBBBBB Nov 06 '23 at 01:53
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    In my opinion, I think that it is a trap to believe that reference books will help you learn a subject. I think it is best to learn from a book that chooses its topics judiciously, and motivates and explains the material well. Then, you can use the reference books as a ... reference ... if you need to quickly remind yourself of a topic. For real analysis, I think Abbott is a good choice for learning (as is Spivak's Calculus), and Rudin is a good reference. – Joe Nov 06 '23 at 01:53
  • Does this answer your question? It's even linked to this question, as are other suggestions. Good book for self study of a First Course in Real Analysis – Ethan Bolker Nov 06 '23 at 01:56
  • @Joe +1, thanks. Sorry I didn't clarify; I meant a good reference book to know once I have understood most of it. I am currently studying with Abbott and Ross so far. And you have just confirmed my supposition that Rudin is the main text for reference. – fiftytwo Nov 06 '23 at 01:58
  • @fiftytwo: Have you had any particular troubles with Abbott? If not, then I would suggest continue reading – I don't think you will find a much better book elsewhere. Personally, I really like Spivak's Calculus, but it might not be the best to start off with if you haven't taken an "intro to proofs" course (or something similar) at your University. – Joe Nov 06 '23 at 02:02
  • @Joe Well, I'm trying to slowly get through the book (I can't complete all the problems). It's a really good book to self study as with Spivak, but once I have finished it, I want to understand analysis ore thoroughly, and I think Rudin's book can do that. – fiftytwo Nov 06 '23 at 02:04
  • @fiftytwo: That's fair. I wouldn't worry about not being able to complete all of the problems. Learning analysis for the first time is hard – it is often people's first exposure to the definition-theorem-proof style of mathematics. It helps to have a teacher to help you along the way. I don't think it is a bad idea to try to read Rudin after finishing Abott, but you might want to try learning some other areas of mathematics first – say linear algebra, abstract algebra, or elementary set theory. – Joe Nov 06 '23 at 02:07
  • @fiftytwo: These areas are not logical prerequisites for Rudin's book, but I do think it will help raise your "mathematical maturity". And you definitely need to have a lot of that to be able to read Rudin. – Joe Nov 06 '23 at 02:08
  • @Joe Thank you for your recommendations! I have studied algebra before, specifically up to rings and fields with Lee and Dummit-Foote. But I see that Analysis is a major subject to learn before I can study differential geometry so I think It would be wise to learn this subject first. Unfortunately I've been denied entrance to university... so its mainly self-study. – fiftytwo Nov 06 '23 at 02:12
  • The analysis you need for differential geometry is only a subset of real analysis. Maybe read Spivak’s Calculus on Manifolds, consulting Hubbard and Hubbard for intuition, then dive into Spivak’s Differential Geometry books or Lee’s books on smooth manifolds etc. – littleO Nov 08 '23 at 06:54

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I recommend you should try Mathematical Analysis by Apostol, it is elementary as well as advanced for real analysis, here you will get multi-dimensional Riemann Integral and Lebesgue Measure with the touch of basic analysis of sequence and series, overall it's really a good book. Thank you.

Albert
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