It is well-known (Liouville's Theorem) that a complex projective manifold X does not admit any non-constant algebraic (or holomorphic) functions. We explain how the collection of all algebraic vector bundles on X and morphisms between them gives rise to a structure (the derived category of X) which serves as a replacement -in many interesting ways - of the L^2 space of functions in analysis. Several results describing the properties of this structure will be explained.