Posted on 2010-11-19, updated at 2012-10-16. By anonymous.
W--y | 1988 | ISBN: 0471608424 | 616 pages | PDF | 21 MB
This two-volume text, originally written in German while Courant was still at Gottingen, is very much better for a serious student than most introductory texts on analysis. Most introductory texts have a flavor of having been written by geniuses for idiots; in this book, Courant treats the student as being his peer in intellect and interest, lacking only knowledge. This makes it an excellent book even for somebody reasonably familiar with the calculus. Although it covers the material from a strictly classical viewpoint, the text and the examples provide enough thinking material to help the student understand the motivation that led to measure theory, Lebesgue-Stieltjes integration, and algebraic topology; the wellsprings of these in classical analysis are seldom explained in modern math courses. So I can recommend it to any senior planning to do graduate work in math, or to any first-year graduate student in math. And of course, it can be well used as a first calculus text for students who are prepared to think and put in effort on the subject.
Courant himself, of course, was a great mathematician, although I don't personally consider him one of the greatest mathematicians of the 20th century; he was a better leader and inspirer of others than a creator of new mathematics. But among other things, he served as David Hilbert's personal assistant for two years, and this gave him superb judgment about what's important and what isn't. This shows throughout the book.
It also helps that the translator into English was E. J. McShane. McShane is less well-known than he perhaps deserves to be, because he was a truly first-rate mathematical researcher (in analysis) himself. This, together with the fact that McShane spent a year or two at Gottingen while Courant was still leading the Mathematics Institute at Gottingen, and came to know Courant well, allowed McShane to translate Courant's text with great understanding of
Courant's way of thinking.
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