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I took the first functional analysis course last semester, though the majority of this was topology, metric spaces and a little on Banach and other more structured spaces at the end. This semesters functional analysis cours eis a 4th year course that continues from the previous one and does more actual functional analysis (Hilbert Spaces etc...).

Alternatively there is the 3rd year differential geometry course which mainly sounds like it is contained in R^n. I've been going to lectures for both and the DG lectures are better presented for me and I also have a comprehensive functional analysis book I was given a year ago as a present - my worry is I don't get the same experience with self study as I do doing assessment.

I probably can't take whichever course I don't do now next year in honours. To give some context courses I'm looking at next year include advanced quantum (introduces relativistic and field theory), general relativity, measure theory and algebraic topology.

Any suggestions on which course I should take? I feel functional analysis is really important, but I haven't really been exposed to much geometry before either...

Thanks for any help!