Background for Graduate Real Analysis I Class

Question

This semester, I have signed up for a graduate Real Analysis I course (really a course in measure theory/Hilbert Spaces/Lebesgue integration) and have thus far attended two lectures. However, from what I see thus far, I feel concerned that my preparedness for this course is inadequate/incomplete. While I have had a standard course in undergraduate real analysis (continuity, differentiation, Riemann integration...) and was briefly introduced to these ideas in $\mathbb{R^n}$, the professor has thrown out there terminology like Jordan Measure, which is encountered in a second undergraduate course in Real Analysis and thus did not have a chance to take as undergrad. The book we are using is E. Stein's "Real Analysis: Measure Theory, Integration, and Hilbert Spaces". The question thus is given the fact that I only had one course in real analysis, is it a good idea to continue through this course? Otherwise, my background in other subjects is strong, I have taken graduate classes in General Topology, say and am comfortable with rigorous proofs at this level.

Answer 1

Functional analysis is the study of topological vector spaces. That is, a vector space with a topology. If you took graduate topology, and did well, you should have the background for the topology aspect. The other aspect is the vector space part. I'm sure you've taken linear algebra, right?

However a one line platitude means nothing when you're asked to prove something "almost everywhere" and find yourself googling the term. :)

You need measure theory. Most functional analysis classes will teach you a minimal amount of measure theory but you need to know it more to do well. Study measure theory on your own. You don't have to get too technical. Just be able to say what a Lebesgue integral is, how it differs from a Riemann integral, and important lemmas like dominated convergence, monotone convergence, etc.

Other than that, if you know point set topology, vector spaces, and have a general idea of what analysis is about, just accept that this is probably the hardest math class you well ever take. I am not exaggerating. First year functional analysis is brutal. Granted math gets harder as you move on, but it won't be taught in classes.