I’m trying to prepare for further study in Analysis and was wondering what advice you all would give. I have read (most of) Abbott’s Understanding Analysis and have started Rudin’s Principles of Mathematical Analysis. I plan to finish Abbott within the next couple of weeks and also Rudin by the middle of June or early July. Would this be adequate preparation for Royden’s Real Analysis or (preferably) Folland’s Real Analysis? Note: I have taken set theory, a 300-level linear algebra course, and will be taking point set topology this upcoming summer.
I say that Folland is preferable because I would, so long as everything goes well, like to take my university’s graduate real analysis sequence, and Folland is the assigned text. I would like to advance to this level because I’m not sure that my school will offer the second semester of their senior-level real analysis sequence – they don’t always.