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!!