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With all of this said, I would suggest taking RA even if it comes too late for the admissions process. I really don't believe a person can be over-prepared in mathematics. The best voices on this subject are those already in programs. Zavera, Ramlau?, Snappy, etc
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I'll agree with this. No such thing as too much math prep. Taking RA, anywhere along the lines will help. Taking
any sort of analysis course will help, or really, any serious proof-based mathematics course. RA is still the most desired because it has the most direct application.
I'm starting in on my fourth week here today, and I use RA, measure, and even my Matrix Analysis, material on just about a daily basis. If nothing else, picking up a
solid foundation on how to write proofs - esp. knowing when you've actually proved something, - seems to be very, very important. And the material is quite important too.
A good grade in Analysis seems to be a signal of three things:
1) being able to write proofs
2) being able to read/write at a high technical & high level
3) knowing analysis
1 and 2 are imperative. 3 is very important.
as far as being able to pick up mathematics when you're in grad school...I don't know how anybody can do it, or do it well. in grad school, you're pretty much expected to learn more than you can really handle - I figure we cover 6-8 chapters a week. that's a struggle in and of itself without trying to pick up a course like RA, which makes even some of the best math majors squirm. Get it done ASAP.