I've been reading up on the finite element method, and the text many people recommend is The Mathematical Theory of Finite Element Methods by Brenner and Scott. As part of the background, the authors recommend (vaguely) "a course in Real Analysis", and then they cite Rudin's "Real and Complex Analysis", Royden, and Folland. Since Ridgway Scott has stated a clear preference for "Big Rudin" (for instance, see this review of Scott's numerical analysis book), it is filtering to the top of my self-study list.
Poking around on Math.SE, I've seen a few people recommend studying abstract algebra before reading "Real and Complex Analysis". So my question is: how much abstract algebra is necessary to understand the first 15 chapters? (Of these, the first 9 or 10 are probably most important.) I ask because:
- I studied the first semester of algebra with Artin several years ago, so I remember (albeit vaguely) what a coset is, etc.
- I do computational mathematics and engineering with a heavy analysis bent, and thus rarely use abstract algebra
- Rudin's text seems to mostly mention ideals, which I've never covered formally, but...
- ...the one introductory functional analysis class (again, years ago) I took also mentioned ideals, and didn't seem to do anything particularly deep with them, beyond showing that other structures were ideals (or left ideals, or right ideals)