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So I recently finished studying Spivak's calculus (excluding the chapter on construction of the real number system) and I'm almost done with Abbott's understanding analysis, what should my next step be? I had a look at Rudin's book and the topics seem so random to me, as in I feel like there is a gap that needs to be filled before diving into Rudin.

So, I'm after some suggestions in regards to my next step, is there a textbook(s) that can bridge the gap between the textbooks I mentioned above and Rudin's book or should I just 'fight' my way through it? Any advice or suggestions will highly be appreciated.

p.s; please keep in mind that this is for self-studying, I'm talking my first 'official' course in real and complex analysis later next year and I want to have a very good grasp on analysis at an undergraduate level ( or even basic grad level) by then.

Here, I'm referring to Real And Complex analysis by Walter Rudin

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    $\begingroup$ "Rudin's book" is an ambiguous reference. Please specify which book by Rudin you're talking about. $\endgroup$ – Henning Makholm Jun 18 '18 at 12:25
  • $\begingroup$ Apologies, it has been edited. $\endgroup$ – Cbb7 Jun 18 '18 at 12:27
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    $\begingroup$ You shouldn't visit Real and Complex Analysis yet. Take a look at Principles first. $\endgroup$ – Allawonder Jun 18 '18 at 12:34
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    $\begingroup$ Ooookkkayyy, now that looks much more inviting. Thank you very much! $\endgroup$ – Cbb7 Jun 18 '18 at 12:37
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    $\begingroup$ For studying Rudin's Principles on your own, be sure to get Bergman's supplementary exercises ... math.berkeley.edu/~gbergman/ug.hndts/m104_Rudin_exs.pdf $\endgroup$ – GEdgar Jun 18 '18 at 13:05
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After Spivak's Calculus, a reasonable next step would be one of:

  1. Calculus, Vol. 2 by Apostol (especially if you need quick acquaintance with multivariable calculus for physics or other reasons)

  2. Mathematical Analysis by Apostol

  3. Principles of Mathematical Analysis by Rudin

  4. Mathematical Analysis by Zorich (perhaps starting partway through Volume 1)

  5. Advanced Calculus by Loomis and Sternberg

Any of the last four would be more than adequate preparation for Rudin's Real and Complex Analysis.

If you have no prior acquaintance with multivariable calculus, but would like to study it at a relatively high level, then Zorich's book might be your best choice. Books 2, 3, and 5 prove the main theorems of multivariable calculus but aren't a good introduction to it. Only books 4 and 5 do a good job on differential forms, Stokes' theorem in a general setting, etc. Only Zorich addresses general topology rather than just metric spaces.

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