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Chapter 1 deals with the problem of the existence of an upper/lower envelopefrom a convex cone or, more generally, a convex set for functions on theprojective limit of vector lattices with values in the completion of theKantorovich space or on the extended real line. Vector, ordinal and topologicaldual interpretations of the existence conditions of such envelope and themethod of its construction are given. Chapter 2 presents applications to theexistence of a nontrivial (pluri)subharmonic and/or (pluri) harmonic minorantfor functions in domains from finite-dimensional real or complex space. Generalapproaches to the problems of nontriviality of weight classes of holomorphicfunctions, to the description of zero (sub)sets for such classes of holomorphicfunctions, to the problem of representation of a meromorphic function as aratio holomorphic functions from a given weight class are indicated.

Order version of the Hahn-Banach theorem, and envelopes. II. Applications to the function theory