TITLE: What is the interface between functional analysis and algebraic geometry?
QUESTION [13 upvotes]: 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?"

REPLY [7 votes]: Have a look at how the Hirzebruch-Riemann-Roch can be deduced as a special case of the Atiyah-Singer index theorem.  The idea is to consider the hodge operator $\overline{\partial} + \overline{\partial}^\ast$ on the bundle of differential forms with values in a given holomorphic vector bundle $V$ over a complex manifold $X$.  This is an elliptic differential operator, and the index theorem says that its Fredholm index is the integral of the product of the Chern class of $V$ and the Todd class of $X$.  On the other hand the Fredholm index is the holomorphic Euler characteristic of $V$ by Hodge theory, yielding HRR.
The proof of the index theorem invariably uses a lot of functional analysis - either analysis of heat kernels, pseudo-differential operator theory, or operator algebras, depending on your preferences.  This provides a foundation for a number of other interactions between algebraic geometry and functional analysis, like generalizations to holomorphic non-commutative spaces or analytic counterparts of Grothendieck-Riemann-Roch.