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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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I'm glad someone mentioned Keith Conrad's notes, as they are excellent.

I would also like to point people towards Tom Weston's webpage. He has expository papers at http://www.math.umass.edu/~weston/ep.html on several topics, including cobordism theory and spectral sequences.

He also has some course notes at http://www.math.umass.edu/~weston/cn.html, including truly excellent book-length notes on introductory algebraic number theory, as well as several dozen illuminating pages on local fields and ideles.

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

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Check out Jean-Pierre Demailly's books on analytic algebraic geometry http://www-fourier.ujf-grenoble.fr/~demailly/books.html.

Here you go the AMS book online webpage http://www.ams.org/online_bks/online_subject.html .

I should also mention the AMS online book webpage collection http://www.ams.org/online_bks/online-books-web.html.

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"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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  • $\begingroup$ I'm currently teaching linear algebra, and using this book. It's ok and as you say has a lot of good problems. The only drawback is that it's arrangement seems a bit confusing to me, and my students. $\endgroup$
    – GMRA
    Commented Oct 28, 2009 at 15:45
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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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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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  • $\begingroup$ Great! Thanks for pointing that out. A pity that the table of contents and index seem to be missing. $\endgroup$
    – Michael Bächtold
    Commented May 25, 2011 at 20:33
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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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  • $\begingroup$ Evens, not Evans $\endgroup$
    – Gerald Edgar
    Commented Oct 22, 2009 at 13:29
  • $\begingroup$ You are correct. $\endgroup$
    – Charles Rezk
    Commented Oct 22, 2009 at 14:56
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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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Sergei Winitzki has an interesting-looking book on the coordinate-free approach to linear algebra online.

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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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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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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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  • $\begingroup$ This link is broken. $\endgroup$
    – David White
    Commented Feb 29 at 1:15
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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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Jim Pitman's Combinatorial Stochastic Processes.

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

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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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The Comptes Rendus de l'Académie des Sciences are available online.

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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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