I think I need a book on pure mathematics to remind of all the things I have forgotten or only dimly remember – partial derivatives, infinite series, various integral forms, Hilbert spaces etc, etc.

What would people recommend?

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Recommending math books is always tricky business — especially for me, since the books I know are ones I used in a previous millenium (and not towards the end of it, either). In any case, it’s a fairly safe bet that you will not find all the topics you listed in one book.

That said, if your goal is to brush up on things you once knew, and you are more concerned with how to use it than with why it is true, I recommend the Schaum’s Outline series. They are economical, easy to read, contain numerous computational examples, and are high quality. (In fact, I had a graduate mathematics class where the instructor used a Schaum’s volume as the sole text.)

If you are looking for a higher level/deeper dive, for Hilbert spaces you could do worse than “Introduction to Hilbert Space” by Halmos (http://www.amazon.com/Introduction-Hilbert-Space-P-Halmos/dp/0828400822), which I used as a graduate student. It is succinct but thorough.

For earlier topics, I had good luck as a student with “Advanced Calculus” by Loomis and Sternberg. It is now out of print, but you can apparently download it free (!) from Sternberg’s web site (http://www.math.harvard.edu/~shlomo/). I also had good luck with “Introduction to Calculus and Analysis Vol. 1” by Courant and John (http://www.amazon.com/Introduction-Calculus-Analysis-Classics-Mathematics/dp/354065058X). For context, bear in mind that I was a math major — I’m not sure an engineer, biologist etc. would have the same reaction. (I long ago gave up trying to imagine what a physicist would think about anything.)

Thanks. I’ll have a look at those. Hilbert spaces not the most important – but interest piqued by biography of Turing.