 In it's most generality, the Hilbert functor is defined for a finite type map of algebraic spaces formula_56 defined over a scheme formula_46.Then, the Hilbert functor is defined asformula_58sendingformula_59This functor is not representable by a scheme, but by an algebraic space.Also, if formula_60, and formula_61 is a finite type map of schemes, their Hilbert functor is represented by an algebraic space.