What is the interface between functional analysis and algebraic geometry?

This is a very open ended curiosity of mine and I would be grateful to hear any comments in this direction. In particular I am interested in functional analysis/algebraic geometry books/papers references which show this bridge from functional analysis into algebraic geometry.

I am not sure if its related but what are the good references for functional analysis on manifolds"? Like how do we characterize the function space based on the domain manifold properties or for specific manifolds like say spheres. (the related things I see are courses like, http://www.math.uiuc.edu/~palbin/Math524.Spring2012/LectureNotesMay1.pdf or http://www.math.harvard.edu/~canzani/math253.html but these seem more about understanding specific differential operators on manifolds rather than the space of functions on a manifold)

Like is there any meaning to wondering, "What is the Hilbert space of functions on a sphere?"

• One important connection between functional analysis and algebraic geometry is Hodge theory, for which you need to understand the Laplacian (elliptic) as well as some related operators. A concise reference are the books of C. Voisin. – M.G. Aug 1 '16 at 16:58
• You might want to consider asking your two questions in separate posts. At least the second part might be considered "too elementary" by some (though it seems okay to me), so math.stackexchange.com is another option. – Nate Eldredge Aug 1 '16 at 17:03
• The interface was, I believe, Alexander Grothendieck ;-) – José Figueroa-O'Farrill Aug 1 '16 at 17:26
• @Jake Why on earth does it come to mind for you? What about Connes's NC(D)G is algebro-geometric? – Yemon Choi Aug 2 '16 at 3:25
• Take a look at the theory of algebraic analysis. Things like Sato's hyperfunctions, microlocal analysis, singular support, etc. A good (albeit terrifying) reference is Kashiwara and Schapira's "Sheaves on Manifolds". – Avi Steiner Oct 22 '17 at 4:17