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?"