Dual representation of superlinear functionals and its applications in function theory. I

We establish conditions under which there is a dual representation of a superlinear functional on a projective limit of vector lattices. These results will enable us, in the second part of this paper, to pose new dual problems for weighted spaces of holomorphic functions of one or several variables defined on a domain in $\mathbb C^n$, namely, the problems of non-triviality of a given space, description of null sets, description of sets of (non-)uniqueness, existence of holomorphic functions of certain classes that play the role of multipliers “suppressing” the growth of a given holomorphic function, and representation of meromorphic functions by quotients of holomorphic functions from a given space.