Complete course of self-study

This is a recapitulation and extension of what we talked about in chat.

Whatever you do, I recommend that you try a variety of areas in order to find out what you like best. Don’t feel obliged to stick to the most common ones, either; for instance, if you find that you’ve a taste for set theory, give it a try.

My own interests are outside the undergraduate mainstream, so in mainstream areas others can probably give better recommendations. I do know that you’re working through Herstein for algebra; although it’s a little old-fashioned, it’s still a fine book, and anyone who can do the harder problems in it is doing well.

You mentioned that you’d prefer books and notes that are freely available. The revised version of Judy Roitman’s Introduction to Modern Set Theory is pretty good and is available here as a PDF. You can also get it from Barnes & Noble for $8.99. Introduction to Set Theory by Hrbacek & Jech is also good, but it’s not freely available (or at least not legitimately so).

I’ve not seen a freely available topology text that I like; in particular, I’m not fond of Morris, Topology Without Tears, though I’ve certainly seen worse. If you’re willing to spend a little and like the idea of a book that proves only the hardest results and leaves the rest to the reader, you could do a lot worse than John Greever’s Theory and Examples of Point-Set Topology. It’s out of print, but Amazon has several very inexpensive used copies. (This book was designed for use in a course taught using the so-called Moore method. It’s excellent for self-study if you have someone available to answer questions if you get stuck, but SE offers exactly that. In the interests of full disclosure I should probably mention that I first learned topology from this book when it was still mimeographed typescript.) If I were to pick a single undergraduate topology book to serve both as a text and a reference, it would probably be Topology, by James Munkres, but I don’t believe that it’s (legitimately) freely available. You might instead consider Stephen Willard, General Topology; it’s at a very slightly higher level than the Munkres, but it’s also well-written, and the Dover edition is very inexpensive.

I can’t speak to its quality, but Robert B. Ash has a first-year graduate algebra text available here; it includes solutions to the exercises, and it introduces some topics not touched by Herstein. He has some other texts available from this page; the algebra ones are more advanced graduate level texts, but the complex analysis text requires only a basic real analysis or advanced calculus course.

This page has links to quite a collection of freely available math books, including several real analysis texts; I’ve not looked at them, so I can’t make any very confident recommendations, but if nothing else there may be some useful ancillary texts there. I will say that this analysis text by Elias Zakon and the companion second volume look pretty decent at first glance. For that matter, the intermediate-level book on number theory by Leo Moser available here looks pretty good, too, apart from having very few exercises. Oh, come to think of it there is one real analysis book that I want to mention: DePree and Swartz, Introduction to Real Analysis, if only for its wonderful introduction to the gauge integral.

Here's a bunch of lesser known material I found really good. It may be a somewhat biased list though, since I tend to like reading "easy" books first to get the main idea, then solving hard problems and moving on to more difficult books later. So, some of this list is less textbook'y and more motivational. It's also a bit biased towards mathematics that is relevant to numerical methods, as that is field of personal interest to me.

Linear Algebra:

• Gilbert Strang's video lectures

Numerical Linear Algebra:

• Trefethen & Bau's book

General techniques for solving mathematical problems:

• How to Solve It
• Terry Tao's book

Abstract Algebra:

• Benedict Gross's video lectures
• Pinter's Book

Convex analysis and optimization:

• Steven Boyd's book
• Steven Boyd's video lectures
• Numerical Optimization by Nocedal and Wright

Real Analysis:

• Francis Su's video lectures

Functional Analysis:

• Kreyszig's book

Mathematical inequalities:

• The Cauchy Schwarz Master Class

Overview of mathematics from Courant:

• What is Mathematics

Topology:

• Munkres book

Mathematical physics:

• Vladimir Arnold's book
• Road to Reality

Numerical/computational mathematics

• Gilbert Strang's video lectures on computational science
• Gilbert Strang's video lectures on mathematical methods
• Wolfgang Bangerth's video lectures on finite element methods

Differential topology/geometry:

• The Shape of Space

Principles of Mathematical Analysis by Rudin is strongly recommended.

Although it may be difficult for you, it will be very impressive if you can read through it. Although I don't know your interests, this book is critical for all students studying mathematics.