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I'm looking for some book to study metric spaces. 2 years ago, used a book called Burkill, as well as using multiple topological concepts, I have also studied the Munkres, Chapters 2,3,4,5,6,9. Anyway, I do not feel as comfortable as well in group theory, rings or general topology )=. So I feel that books such as Hungerford, Robinson Group, Atiyah Mcdonald. They have been very important in my formation. But I have not had that feeling with any book of analysis (metric spaces), I feel my training is still very basic. What book would you recommend to study analysis?

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5 Answers

Rudin, Principles of Mathematical Analysis.

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A good book for metric spaces specifically would be Ó Searcóid's Metric Spaces. However, note that while metric spaces play an important role in real analysis, the study of metric spaces is by no means the same thing as real analysis.

A good book for real analysis would be Kolmogorov and Fomin's Introductory Real Analysis.

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There is a wonderful short book by Kaplansky called Set Theory and Metric Spaces.

How good is it? I was able to read it at the beginning of my undergraduate career, and in the intervening years of undergraduate study, graduate study and then mathematical research, I have only ever turned to other books on either of these subjects out of idle curiosity: everything I have ever needed to know is in Kaplansky's text.

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I'll second the recommendation on Kaplansky's text. It's very easy to read and is inexpensive; a perfect first book on Metric Spaces. –  ItsNotObvious Dec 8 '11 at 17:00
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Kaplansky was one of the great master teachers at the University of Chicago in it's heyday and it's very sad most of his lecture notes are out of print and very expensive. –  Mathemagician1234 Jan 21 '12 at 0:10
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Dieudonné's Foundations of modern analysis Chapter 3 is a very thorough treatment of metric spaces. It's a bit dry, but perfect as reference.

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