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I often use the internet to find resources for learning new mathematics and due to an explosion in online activity, there is always plenty to find. Many of these turn out to be somewhat unreadable because of writing quality, organization or presentation.

I recently found out that "The Elements of Statistical Learning' by Hastie, Tibshirani and Friedman was available free online: http://www-stat.stanford.edu/~tibs/ElemStatLearn/ . It is a really well written book at a high technical level. Moreover, this is the second edition which means the book has already gone through quite a few levels of editing.

I was quite amazed to see a resource like this available free online.

Now, my question is, are there more resources like this? Are there free mathematics books that have it all: well-written, well-illustrated, properly typeset and so on?

Now, on the one hand, I have been saying 'book' but I am sure that good mathematical writing online is not limited to just books. On the other hand, I definitely don't mean the typical journal article. It's hard to come up with good criteria on this score, but I am talking about writing that is reasonably lengthy, addresses several topics and whose purpose is essentially pedagogical.

If so, I'd love to hear about them. Please suggest just one resource per comment so we can vote them up and provide a link!

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There is a whole database of freely available books at e-booksdirectory.com/mathematics.php; I'm making this a comment since it's not one specific volume. –  Akhil Mathew Oct 21 '09 at 21:26
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There are several links, both to specific free books and to databases, in a related question: mathoverflow.net/questions/761?sort=votes#sort-top –  Qiaochu Yuan Oct 21 '09 at 21:30
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A list which I don't think is mentioned in the other question is here: people.math.gatech.edu/~cain/textbooks/onlinebooks.html –  Qiaochu Yuan Oct 21 '09 at 21:37
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Please, add them to mathonline.andreaferretti.it too. :-) –  Andrea Ferretti Nov 19 '10 at 14:47
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54 Answers 54

"Linear Algebra" by Jim Hefferon has been online for a while and it was what I used to teach myself linear algebra. It's very well written with tons of great practice problems and interesting asides. It is a little less advanced than any of the other books listed so far, but it's still a great read. Plus, it's open source (You can download the LaTeX for the book from the website).

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And another great source of the lecture notes and stuff is the MIT OpenCourseWare, in particular the math section.

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A draft of Albert Marden's Outer circles: an introduction to hyperbolic 3-manifolds is online, on his website:

http://www.math.umn.edu/~am/book/outercircles.pdf

Edward Nelson's Radically elementary probability theory is also online, on his website:

http://www.math.princeton.edu/~nelson/books/rept.pdf

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The Stacks Project http://math.columbia.edu/algebraic_geometry/stacks-git/

If I ever wonder about something, I can pretty much count on it to be in there. Remember to use a PDF viewer with hyperlinks and back/next buttons.

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Exterior Differential Systems by Bryant, Chern, Gardner, Goldschmidt, and Griffiths is available through MSRI (and is sadly out of print at the moment).

http://library.msri.org/books/masterlist.html

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'The Elements of Statistical Learning' by Hastie, Tibshirani and Friedman http://www-stat.stanford.edu/~tibs/ElemStatLearn/

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Len Evens has a couple of online textbooks: a text on abstract algebra, and a linear algebra text.

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Martin J Osborne and Ariel Rubinstein's book "A course in Game theory" is available here.

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The Caltechbook service at Caltech offers a number of math books for free here, including some very good (IMHO) books by Jerry Marsden et al.

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Jerome Keisler's Elementary Calculus. This book uses infinitesimals explicitly, and also in a logically rigorous way, without getting too advanced for first-year undergraduates.

Later edit: http://math.wisc.edu/~keisler/calc.html

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Richard P. Stanley's Enumerative Combinatorics, volume 1, second edition is available at http://www-math.mit.edu/~rstan/ec/ec1/ .

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Jim Pitman's Combinatorial Stochastic Processes.

Later edit: http://works.bepress.com/jim_pitman/1

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A url would be nice. –  Robin Chapman Nov 20 '10 at 8:51
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Book by David Levin, Yuval Peres and Elizabeth Wilmor on Markov chain theory and mixing times. http://pages.uoregon.edu/dlevin/MARKOV/markovmixing.pdf. It quickly takes someone with basic knowledge in probability and linear algebra into the heart of current research.

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Gerald Teschl books

  1. Textbook Ordinary Differential Equations and Dynamical Systems
  2. Textbook Mathematical Methods in Quantum Mechanics; With Applications to Schrödinger Operators

can be found at http://www.mat.univie.ac.at/~gerald/

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The website of the Leibniz award offers a free online collection: link, and there is the preprint server of the IHES.

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Within the framework of the project retro.seals.ch, scientific journals are retrodigitized and made available via internet. The project contains the following mathematical journals:

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As with so many things, "There's a reddit for that": http://www.reddit.com/r/mathbooks. It's a mixed bag—much like searching for math books in a non-specialist bookstore, one gets the elementary mixed up with the sophisticated—but there are some gems there.

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Roland Speicher has some nice introductory material for Free Probability (mini course, survey articles etc.) All available at http://www.mast.queensu.ca/~speicher/survey.html

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Basic Concepts of Enriched Category Theory by G.M. Kelly was originally published by Cambridge University Press in 1982 but is now available online: http://www.tac.mta.ca/tac/reprints/articles/10/tr10abs.html

My understanding is that it is the canonical reference for enriched category theory (and was written by the pioneer of the field).

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