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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.

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    $\begingroup$ 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? $\endgroup$ – Hugo C Botós Mar 8 '20 at 14:08
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    $\begingroup$ 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. $\endgroup$ – Hugo C Botós Mar 8 '20 at 14:12
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    $\begingroup$ @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? $\endgroup$ – user743574 Mar 8 '20 at 14:15
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    $\begingroup$ @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. $\endgroup$ – Hugo C Botós Mar 8 '20 at 14:21
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    $\begingroup$ Does this answer your question? Selecting the Real Analysis Textbooks $\endgroup$ – Intellectually disabled Dec 22 '20 at 3:16
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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

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  • $\begingroup$ 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. $\endgroup$ – Chris Leary Mar 26 '20 at 19:02
  • $\begingroup$ 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. $\endgroup$ – user743574 Mar 27 '20 at 4:12
  • $\begingroup$ 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. $\endgroup$ – user743574 Mar 27 '20 at 4:18
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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.

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