The Extensive Properties Of A Mapping In Ordered Vector Spaces

The classical Hahn-Banach extension theorem indicates that: if T is a linear operator from a subspace G of a vector space X into a Dedekind complete Riesz space Y and T is dominated by a sublinear operator P defined on X ,then T will be extend to a linear operator T|- from X into Y ,at the same time, dominated by P. The conclusion can be generalized to positive operator, so we have the classical Kantorovich extension theorem: every positive operators from a majorizing vector subspace of Riesz space into a Dedekind complete Riesz space always has a positive extension. The result was generalized from a paper which writed by Buskes G.J.H.M and van Rooij A.C.M. They week Riesz space to directed partially ordered vector spaces. It proves that if X is a directed partially ordered vector space, and Y is a Dedekind complete Riesz space, then the weak lattice-homomorphism from a majorizing vector subspace G of X into Y can be extened to a weak lattice-homomorphism all of X into Y. Follow the way of thinking, we will discuss the Kantorovich-type theorem concering the extension of a positive projection from a majorizing vector subspace G of the Dedekind complete partially ordered vector space X can be extened to a positive projection defined on all of X .One of the important generalize of Hahn-Banach extension theorem is to the need of the applications, for example, the optimization problems, generalize the single-valued mapping to the more-valued mapping. So the discuss of the extension problem about the more-valued mapping has important value. We call f is a more-valued mapping from X into Y, if for each x∈X, there are more element y₁, y₂,…∈Y, such that f(x) = y₁ and f(x) = y₂,…. If we take all y₁,y₂,…as a set, then we get a especially mapping of more-valued mappmg, set-valued mapping. In this paper, we discuss the extension theorem concering the set-valued mapping from a vector space into a Dedekind complete partially ordered vector space which the order is induced by cone K. And when the vector space endowed with a locally full topological, become a ordered topological vector space, the extension problem of the continue set-valued mapping.The classical Hahn-Banach extension theorem, Kantorovich extension theorem and most of it's generalization have a very strong need that the value space must be a Dedekind complete vector space. We also show that a extension theorem concering set-valued mapping that the value space is a partially ordered vector space which the order is induced by cone K but not a Dedeking complete vector space.