[math_phd_to_dev]

I think that those who are down-voting people suggesting Rudin are missing the forest for the trees. The level is not as low as Feynman's lectures, but if you want a text that has the hallmarks of a true master at work, Rudin fits the bill. I personally always felt that what made Feynman's lectures what they are were, was that the man actually understood the subjects on such a deep level that his mind was able to hone in on those little simple thought experiments and ways of looking at things that gave you huge insight into what he was lecturing on. Rudin comes at analysis with the same level of understanding and that's why Real & Complex and Baby Rudin are still the gold standard for analysis texts. It is impossible to convey the sense of elegance that mathematicians speak about until you've seen it and been blown away by it. Rudin will do that and leave you speechless, he's the master. I can tell you that my first encounter with Real & Complex leaving me thinking that Rudin was on a fundamentally higher plane than any author I had read before, it was like watching a magician. If you're looking for a real Feynman in mathematics, IMHO Rudin is your man.

[wbhart]

When I did analysis at university, I read Rudin instead of the recommended text. When the exam came around, I tried to generalise all the problems on the exam and gave answers which did not impress the person marking it, as they looked nothing like the standard solutions. As a result, I very nearly failed analysis. I wasn't particularly upset, as I used to consider my exam grade as a reflection of how well the course was taught, rather than a grading of my own comprehension. This is not a level of arrogance I would recommend to students looking to pass their exams.