Introduction
Never before it was so easy to share knowledge: currently it takes moments and costs approximately nothing to send a book copy to the other side of the planet. You can just get a lot of books on any topic, skim those and look for suitable ones, or pick random ones to read, or find the ones you need – better than any offline library, not only in that you can do it anytime and from any computer connected to the internet.
Unfortunately, computers are not always connected to the internet, search engines tend to throw fraud and commercial sites instead of books, some governments blacklist libraries, and some accuse their citizens of digital piracy and threaten with prison rape. Not a big deal, but not particularly convenient; better to know where the books are.
Project Gutenberg
Project Gutenberg is a nice library, particularly for recreational reading: it is all legal, so books get there when their copyright expires; most of useful information gets outdated by that time. For offline reading (as well as for backup purposes), it provides CD images, though not all the books are there.
There are somewhat similar libraries, such as The Online Books Page, but usually it’s even harder to find anything useful there.
Internet Archive
Internet Archive provides tons of texts, including books, among other cool things. Its mission:
Most societies place importance on preserving artifacts of their culture and heritage. Without such artifacts, civilization has no memory and no mechanism to learn from its successes and failures. Our culture now produces more and more artifacts in digital form. The Archive’s mission is to help preserve those artifacts and create an Internet library for researchers, historians, and scholars. The Archive collaborates with institutions including the Library of Congress and the Smithsonian.
Wikibooks
Wikibooks is a very promising project, yet there is very few finished books by now. Since even those are not guaranteed to be the best for you, it is not the most useful library for now, but hopefully it will be in the future.
Wikipedia
Wikipedia is not a library, yet its printed version would consist of a couple of thousands of volumes, which would make a small library. And it’s a great encyclopedia that can be downloaded – to be prepared for a zombie apocalypse, censorship of Wikipedia, internet connection issues, or simply to have lower latency.
Papers and collections
Open access is a particularly important thing, which provides free access to important research results to everyone. Those are not educational books, of course, and one should be familiar with a field in order to read those, yet there is much more of exciting and modern things in papers than there is in educational books – since papers are basically the sources from which those books are assembled. There is a list of open access journals in Wikipedia.
Unfortunately, some papers are behind a paywall (most notably, JSTOR sells them), though there is arXiv.org, where papers are freely available, as well as the open access journals mentioned above, various university websites, and researchers’ homepages.
There also are topical collections of papers and free books, either links or file collections. Look them up on github.
Less legal (but substantially larger) collections include Sci-Hub and LibGen.
BitTorrent trackers
The Pirate Bay (thepiratebay.org), as well as torrent search engines, may help you to find many books; often it is easier to find a specific book there, than by using a general-purpose web search engine.
Unfortunately, they are getting raided by law enforcement agencies, so this section is hard to keep in a good shape.
Offline libraries
The thing about offline libraries is that you can pick, skim, and read anything, anytime, and for free; they can also be written onto a device that fits into your pocket. Unfortunately, it is too good to be legal, so we won’t risk to provide the links or hashes. But here is a few names of mathematical libraries:
The MYRIAD Mathematical Library: around 11 GB of pure (and applied) awesomeness.-- mit math: 2 GB of that.MATH MATHEMATICS BOOK COLLECTION: almost 10 GB.
Tor network libraries
Galaxy2 bookmarks contain a “library” section, and The Hidden Wiki has a page for libraries, and there is plenty of other directories. But while one may think that anonymity leads to great libraries, in fact it is not necessarily the case. Yet there are nice ones, Clockwise Libraries in particular.