I'm wondering whether there exists some geometrical theories of functional spaces. I mean; function spaces ($L^p$ spaces for example) are called topological vector spaces (TVS). I'm interested in whether there are some geometrical difference between each functional space or TVS (local/global convexness? Alexandrov curvature?)(I found vector space is always global convex, sorry! in 11/Dec '18).
And I'd like to take a look at some references about this kind of topic.
Anything will help, thank you!
I happen to have a small list of some geometric theory of Banach spaces, I hope you might find some of these results interesting.
"Topological equivalence of all separable Banach spaces": Kadec proved that all separable Banach spaces are homeomorphic.
"On the nonexistence of uniform homeomorphisms between $L^p$ spaces": Enflo proved the result for $1\le p,q \le 2.$
"On nonlinear projections in Banach spaces": Lindenstrauss proved that for $p,q\ge 1$, if $\max(p,q) > 2$ then $L^p$ and $L^q$ are not uniformly homeomorphic.
Maurey's "Type, cotype and K-convexity, in Handbook of the Geometry of Banach Spaces, Vol. 2": Type & Cotype is the method that is used nowadays to show that there's no isomorphism between different $L^p$ spaces.