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I'm a sophomore and I'm taking my introductory real analysis course this summer.

What I'm not sure about is the proper textbook to use in order to learn it properly, at MIT they jump to Rudin at once, they don't bother studying any lowered down books before it so I know that I do not have to study mathematical analysis from lowered down books before I jump to the real deal.

Courses I have taken so far:

  • Calculus (although I'm bad at the multivariable calculus I still can revise anything I need whenever needed).
  • Elementary Set Theory
  • Abstract Algebra (Group Theory)

and I'm taking topology this semester as well. I know nothing of linear algebra though but I do not think its really necessary. So what do you think is the proper textbook to use?

Is it Rudin/Pugh/Apostol or Do I have to learn from books like Abbott/Kenneth Ross/Bartle first? I have already studied few sections from Bartle regarding Continuity and Differentiation and I find the exposure to the material quite weak, doesn't matter if I do all the problems I still do not taste the beauty of analysis from this *hit textbook because all I can find in it is just a bunch of definitions with no explanation on why are they there or what is the point of having them and the problems are just direct random applications so if I do them I still feel like I learned absolutely purely nothing new just calculus written in a different way really.

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  • $\begingroup$ How did your class turnout? What book did you guys use? $\endgroup$ – smokeypeat Jul 17 '17 at 0:13
  • $\begingroup$ "I know nothing of linear algebra though but I do not think it is really necessary." It is crucial to multivariable calculus/analysis. Side note and advice given to me, I don't think it's even possible to know too much linear algebra. $\endgroup$ – qbert Apr 26 '18 at 20:14
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Try Spivak Calculus. It is well written and has challenging exercises.

A solution manual is also available.

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    $\begingroup$ Spivak is only helpful as an additional reading.. its not considered an actual standalone analysis book. $\endgroup$ – Omar Qasem Apr 26 '16 at 8:11
  • $\begingroup$ @OmarQasem why do you say that? Also what is your definition of "Analysis." $\endgroup$ – qbert Apr 26 '18 at 20:11
  • $\begingroup$ @OmarQasem I ask because from the books you are quoting, it seems you mostly want a rigorous treatment of calculus. Spivak does a good job doing that in the single variate case (and is a good precursor to Rudin). $\endgroup$ – qbert Apr 26 '18 at 20:16
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Read Abbott's Understanding Analysis. You won't regret it.

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