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What's the best book to study analysis after finishing Spivak's Calculus? I thought about Rudin's Principles of Mathematical Analysis(Which I guess would be much boring for me, but I don't have it so I can't tell if my supposition is right or not), Stein and Shakarchi's Real Analysis, Terence Tao's Analysis(if you recommend it, tell me which edition is the 'right', please) or Pugh's Real Mathematical Analysis?

Additional info about my purpose: - I tend to seek elegance in proofs. - I want to grasp concepts the most deeply possible. - I don't like books that just jump steps without clear explanation, but I don't like books that are boring(books that focus too much in rigor, in the steps, in the "you can prove it this way". I like rigor and it's what - - - I'm seeking, but sometimes authors make it boring. I don't know if I made me intelligible). - However, I want books that make me try to 'rediscover the subject'; mean, books that make me think hard on the subject even before he explains the matter. - Books with super hard exercises are welcome.

user743574
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  • Pugh‘s book is one of my favorite analysis books. :) – Qi Zhu Mar 08 '20 at 13:22
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    Well, if you already read Spivak's calculus book, then you already have a good understanding of analysis. I think you would be bored reading Rudin's book. Why not try something more challenging like measure theory, functional analysis, Fourier analysis, etc? – Hugo Mar 08 '20 at 14:08
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    By the way, the four books by Stein on Analysis are just wonderful, I recommend them strongly. I have to add that there is a order to this books, the Real Analysis is the third one and it's essentially about measure theory. The order they follow is:
    1. Fourier Analysis: An Introduction.
    2. Complex Analysis.
    3. Real Analysis: Measure Theory, Integration, and Hilbert Spaces.
    4. Functional Analysis: Introduction to Further Topics in Analysis.
     – Hugo Mar 08 '20 at 14:12
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    @Hugocito I would like to start Fourier Analysis but I don't know if I'm prepared. I don't know what are the prerequisites. Do you think Stein's series are good for me, knowing the info I gave above? – user743574 Mar 08 '20 at 14:15
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    @Joãofodão Yes. I believe you are essentially prepared to read "Fourier Analysis: An Introduction." by Stein and Shakarchi. On latter chapters, you will need to know how to integrate on several variables, but you can learn that on your spare time. – Hugo Mar 08 '20 at 14:21
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    And I have to add it is a very beautiful book. I learned a lot from it. – Hugo Mar 08 '20 at 14:22
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    @Hugocito I forgot to ask about exercises. Are there really hard exercises? And does the book make you think very hard on the topics? – user743574 Mar 08 '20 at 15:29
  • @Joãofodão Well, I don't remember, I read it so many years ago. But I suppose there are hard problems there, but you can always ask us for help here. Ah, and there is another recommendation you may enjoy. Take a look at Mathematical Analysis by Vladimir A. Zorich, it covers a lot of analysis, and I'm sure it will not bore you. Furthermore, if you enjoy it, there are hard exercises there. The interesting aspect of this book is that it's written in a very russian style, whatever that means. – Hugo Mar 08 '20 at 15:54
  • @Hugocito Doesn't Zorich cover almost the same thing Spivak already covered in his book, "Calculus"? Ok, it may sound absurd, since Spivak is a introduction to Analysis but not a Analysis text, but I got an idea that Zorich's Volume 1 doesn't cover many more things. Probably, just probably, I'm completely wrong, but I want to have sure that he covers really very good aspects of Analysis beyond Spivak. – user743574 Mar 08 '20 at 16:01
  • You are correct. But there is also the volume 2. – Hugo Mar 08 '20 at 16:03
  • @Hugocito So, is it better to go rightly into Volume 2 after finishing Spivak? – user743574 Mar 08 '20 at 16:05
  • Let us continue this discussion in chat. – Hugo Mar 08 '20 at 16:07
  • Or would it be even better to buy Spivak's Calculus on Manifolds instead? I saw many people saying Calculus on Manifolds is a very though book and I fear that I won't like it because I got a impression that he doesn't teach many things: I saw some people saying he emphasizes too much Stokes Theorem – user743574 Mar 08 '20 at 16:07
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    Does this answer your question? Selecting the Real Analysis Textbooks –  Dec 22 '20 at 03:16

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Well, if you have to choose between the textbooks that you indicated, I'd go with Rudin. But here are some extra thoughts. Pugh's textbook has a topological flavor and is great if you intend to delve further into modern dynamical systems; many (I mean, a great many) exercises may be hard to do, even if you think that you are "fodão". Tao's textbook is too wordy, I can't cope with it. Stein and Shakarchi's text is a mixed bag, but with a nice mix; maybe it's just what you want. An alternative to all these textbooks that in the end worked for me was T. M. Apostol, Mathematical Analysis, 2nd ed. Now if you want to really learn measure and integration theory, you can't go wrong with A. E. Taylor, General Theory of Functions and Integration. #ficadica

Jotazuma
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  • Rudin and Apostol have been around for a long time, and have stood the test of time. They have been valued references for me throughout my career (I've been around a while, too). Newer texts are inevitable, no matter the subject. I'm not sure how many improve on the classics. – Chris Leary Mar 26 '20 at 19:02
  • Thanks for suggestions. I'll finish Calculus and I decide it after. Probably I'll go with Rudin. I just love the way you tell "#ficadica" ahah. The "fodão" part didn't haven't the intention of being Interpretation that way. It's just how older guys say in my country. Here, we call it "calão" that would mean something like extremely informal language and in some contexts may be offensive. Now I wonder if I should change the name. It was written with no offensive intentions, but it may appear for some people. – user743574 Mar 27 '20 at 04:12
  • I would like to remember that I'm in the first book by Spivak, not the book of multivariable calculus. Taking this in account, I don't if it's better to start by studying the beginning of Analysis or just continue learning the "multivariable calculus" without the emphasis of Analysis in theory. – user743574 Mar 27 '20 at 04:18
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i think real analysis is very tricky subject, and i suggest you some books,such as zorich.here is link http://instructor.sdu.edu.kz/~verbovskiy/Math%20Analysis%201/Zorich1_en.pdf. it covers every aspect of mathematical analysis, that you need for mathematical career.My suggestion would be to not afraid of those tough symbols, because early or lately you will be reading research papers, so it would be better if you get used to it now.in general there is no best book, so find one that works for you.i also think that rudin is ok, just push yourself and study those books that i recommended.happy learning.

dato nefaridze
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