Topics in Analysis every young mathematician should know
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Royden's book is the best book on analysis that I have read and worked through. There are authors that cover a subject and there are those who uncover it. Professor Royden is definitely from the latter category. The book starts off with a quick introduction to the real number system, and various misccellaneous facts about real functions that aren't usually to be found in a typical real analysis text, and that cannot be generally found in a typical undergraduate analysis course at a British University. The book then proceeds with an introduction to Lebesgue measure. The exposition in this part of the book is very leisurely, and time is taken to cover the very important special examples of the non-measurable set, and the Cantor function. The only regret I have is that the author mentions Littlewood's three principles, but leaves them there. He could have spent a little more time on these. The second part of the book is a very (what I felt to be) quick paced introduction to topology. Again the exposition is helped in a major way by the very careful selection of problems, which do help to illuminate this section much more than the exposition itself would have been able to do on its own. The third section, which focuses on abstract measure theory is an inspiration. This part completes the very particular discussion of the earlier introduction of Lebesgue measure, rounding it out, and giving the reader a real insight and understanding of measure theory. The book has, however, begun to show its datedness. However, it is a perfect size, not being a dauntingly large tome designed to scare off any readers but the most determined, but at the same time carrying enough detail to provide a solid foundation in analysis. I just wish there were a few more examples, to illustrate the theory. The problems, which have been mentioned above more than once are very well thought out, and the choice couldn't be better.
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