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Although the book Principles of Mathematical Analysis by Walter Rudin, 3rd edition, is a very useful textbook on the subject that it deals with, I sometimes feel that the results stated are sometimes not as general as they can be. For example, there are several results on compact metric spaces (in Chapters 2, 3, and 4) that are valid for any Hausdorff topological spaces; similarly, the concept of uniform convergence (discussed in Chap. 7) can be treated in the context of an arbitrary normed space, or even an arbitrary topological (vector) space! Am I right? If so, then my question is as follows:

Is (or are) there any text (or texts) that cover the same material as has been covered in Baby Rudin but with sufficient more generality?

In particular, I'm a bit discouraged when Rudin restricts the discussion to complex-valued functions in Chap. 7!!

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    $\begingroup$ According to me, that book does pay attention to appropriate levels of generality. When more generality than the one given in the text is available, there are usually several exercises to treat it. $\endgroup$ – Giuseppe Negro Apr 2 '18 at 10:06
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    $\begingroup$ @GiuseppeNegro but Rudin doesn't mention topoligical or normed spaces at all. $\endgroup$ – Saaqib Mahmood Apr 2 '18 at 10:11
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    $\begingroup$ Normed spaces are not needed for that material, the author has another book on functional analysis. Topological spaces could be introduced, but the only benefit would be in the second chapter, in which a couple of theorems could be mildly generalized. (This generalization would come with the significant cost of having to introduce a whole lot of definitions). Moreover, the author has a book on real and complex analysis that does treat topological spaces, especially in the context of Radon measures. $\endgroup$ – Giuseppe Negro Apr 2 '18 at 10:12
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    $\begingroup$ My point is that Rudin already chose a high level of generality, and more importantly, a good balance of generality and clarity. Anyway, if you want a more abstract approach, try opening Hewitt and Stromberg's book: springer.com/gp/book/9780387901381 $\endgroup$ – Giuseppe Negro Apr 2 '18 at 10:18
  • $\begingroup$ try the books of Amann and Escher, or the books of Zorich, or the books of Shakarchi. Also the books of Garling. $\endgroup$ – Masacroso Apr 2 '18 at 11:01

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