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I’ve taken a few courses in advanced calculus and real analysis (adv calc 1, metric spaces, normed spaces, lebesgue integration), but I’m realizing that my advanced calculus/R^n analysis is not as strong as I’d like it to be.

So here’s my question: If I’m planning on spending this summer (4 months) reviewing advanced calculus/R^n analysis, what books/resources would you recommend?

I haven’t seen this question posted on here before, so I don’t think (hope) it’s not duplicate. I’m looking for books/resources which (1) give good exercises, (2) don’t shy away from the more difficult theorems, since I’ve had experience with analysis before, (3) will be at least fairly readable (at least as readable as baby Rudin), & (4) cover at least some of important topology of R^n. (5) Fourier series would be nice but not necessary.

Thank you.

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  • $\begingroup$ You may want to clarify what topics are covered in your "adv calc 1". If you don't want to write down a long list, it may be helpful to mention at least what textbooks you were using for your advanced calculus and/or analysis courses. $\endgroup$ – user1551 Apr 10 at 18:04
  • $\begingroup$ Advanced Calc 1: We used Abbott’s “Understanding Analysis” and covered chapters 1-5, part of 6, and all of 7. Analysis 1: analysis and topology of metric spaces, spaces of continuous functions, approximation of continuous functions (professor’s notes). Analysis 2: normed spaces, L_p and l_p spaces, lebesgue integration (professor’s notes). $\endgroup$ – CategoricalImperitive Apr 10 at 18:49
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I'm glad of your standards of readability, because Rudin, in my opinion, is hardly readable.:)

My recommendation, which seems to match your criteria, is Zorich.

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