The 3 of us are taking Real Analysis this year, and our professor has his own book online, “A Companion to Real Analysis.” It’s dense, contains no proofs, and is generally intense. But, those are good things. I wouldn’t use it by itself, but it provides you a great linear progression, moving from basic set theory, functions, and order, then into Algebra, Vector Spaces, Topology, Continuity, and Normed Linear Spaces. Clearly it’s not complete (he’s constantly updating), as it is only to be used in the first part of a year long course. But, it’s proven to be a solid, and rigorous, introduction to the topic.

[It also contains a digression on Categories, but if you have no prior knowledge of them, you’re libel to be lost for a while.]

Quote:

Most of (real) analysis is a study of the interplay of algebra and topology. Thus far in these notes there has been quite a bit of algebra, but no topology. Now we begin to remedy this unhappy situation.

-Saij

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