Advanced Calculus

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1. GrothendieckApr 2014
   SICP doesn't really cover very much in the grand scheme of CS, so "like SICP" might mean ...

   ... something covering "how to do math". This might be an introductory real analysis or linear algebra book, or (alas, I haven't read much in this area) you could do no wrong looking for books on problem solving, contests, and inequalities, say by Polya or Andreescu.

   ... a higher-level view of day-to-day mathematical practice. There probably isn't one book for this, but I'd recommend Loomis and Sternberg's Advanced Calculus as a summation of linear algebra and calculus on manifolds; you'd also need to read on complex and functional analysis, algebra (Lang and the unofficial companion volume?), topology ...

   ... a broad but shallow introduction to several fields and applications unified but a common underlying approach (abstraction and programming language design). I'd recommend Geroch's Mathematical physics, which, as the name implies, studies algebra, algebraic topology, and functional analysis through the common lens of category theory.

   ... a somewhat quirky book on foundations? You could look for a book on naive or axiomatic set theory, categories, or type theory.

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2. HilbertSpaceNov 2011
   Glad you learned some math in college.

   Commonly in K-12, the best 'math' taught is plane geometry because there, at least when it's a theorem proving course instead of paper cutouts, which sometimes happens, can see in clear terms the roles of the big three -- definitions, theorems, and proofs. You can also see the role of one more -- intuition, especially its best form, geometric intuition. Right: Intuition doesn't prove anything, but it can be one of the best ways to guess what is true and how to prove it. For more, eventually you can get a useful intuitive feeling for a topic.

   There's

   http://www.american.com/archive/2008/march-april-magazine-co...

   which at face value is supposed to be about women in math but describes Harvard's Math 55. At least at one time for that course the three main texts were:

   Paul R. Halmos, 'Finite-Dimensional Vector Spaces, Second Edition', D. Van Nostrand Company, Inc., Princeton, New Jersey.

   Walter Rudin, 'Principles of Mathematical Analysis, Third Edition', McGraw-Hill.

   Michael Spivak, 'Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus', W. A. Benjamin, New York.

   Working successfully through those three is quite sufficient to understand proof-based college math!

   Those three are all old; in particular Halmos wrote the first edition of his book in 1942 when he was an assistant to von Neumann at the Institute for Advanced Study.

   I had Rudin's book in college but later rushed to work carefully through both Halmos and Spivak ASAP after college.

   Instead of Spivak, I preferred:

   Wendell H. Fleming, 'Functions of Several Variables', Addison-Wesley, Reading, Massachusetts.

   Might also consider:

   Lynn H. Loomis and Shlomo Sternberg, 'Advanced Calculus', ISBN 0-201-04305-X, Addison-Wesley, Reading, Massachusetts.

   For exterior algebra, now can get in English:

   Henri Cartan, 'Differential Forms', ISBN 0-486-45010-4, Dover, Mineola, NY.

   Since mentioned Halmos and since this thread is about probability and statistics, should mention that Halmos was one of the best in those topics in the US in the 20th century.

   Halmos was a student of J. Doob at University of Illinois as in:

   J. L. Doob, 'Stochastic Processes', John Wiley and Sons, New York, 1953.

   and has a very nice start on probability in:

   Paul R. Halmos, 'Measure Theory', D. Van Nostrand Company, Inc., Princeton, NJ, 1950.

   Halmos also wrote:

   Paul R. Halmos, "The Theory of Unbiased Estimation", 'Annals of Mathematical Statistics', Volume 17, Number 1, pages 34-43, 1946.

   and also the crucial:

   Paul R. Halmos and L. J. Savage, "Application of the Radon-Nikodym Theorem to the Theory of Sufficient Statistics", Annals of Mathematical Statistics, Volume 20, Number 2, 225-241, 1949.

   Yes 'Finite-Dimensional Vector Spaces' is really a finite dimensional introduction to Hilbert space which mostly have to attribute to von Neumann (who once reminded Hilbert what it was).

   The set of all real valued random variables X such that E[X^2] is finite forms a Hilbert space. The amazing part is completeness, and there is a proof in:

   Walter Rudin, 'Real and Complex Analysis', ISBN 07-054232-5, McGraw-Hill, New York.

   which also has a nice chapter on Hilbert space.

   Yes, having professors write their own books is now more common and can make a course more efficient for the students. It was long the case that a student had to copy the 'text' off the board or just take notes and turn them into a text. Now with TeX and LaTeX, PDF, and the Internet, finally the word whacking for the math can often be less work than the math!

   Still, it will be difficult to improve on some of the best texts, e.g., Halmos. Rudin went through at least three editions of his 'Principles', and the level of polish started high and increased. A good book is actually NOT easy to write.

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