It would be very handy to know about function spaces, distributions and Fourier analysis. It looks like Rudin's Functional Analysis covers these things, but I do not yet have the foundation for it. (see last paragraph)
For someone who has the median knowledge from an "introductory real analysis class," how do you build through these topics?
One option would be to work through 'Real & Complex Analysis' by Rudin before his 'Functional Analysis,' but I am hesitant to commit (dollars) because I have been told he uses 'magic tricks' for the sake brevity while obscuring understanding. Are these people just whiners? The internet is a great supplementary text.
In my sequence we partially worked through Bartle/Sherbert, then the awful pink book by Marsden/Hoffman, if that matters. We went from field axioms & completeness of the reals up through a few simple proofs involving total derivatives and the integral that uses Darboux sums. We avoided discussing actual measure theory by defining a weaker 'volume in $\mathbb{R}^n$'.
Best Answer
Kreyszig's book Introductory Functional Analysis with Applications is a classic that might be good at your level.
I like the way the fundamentals of functional analysis treated in this set of lecture notes.
For more a more advanced book, consider Conway's A Course in Functional Analysis.
It's also useful to really have a solid intuitive understanding of linear algebra and topology. Functional analysis is basically infinite dimensional linear algebra, if you interpret "linear algebra" broadly enough.