The Extension Proposition Of A Mapping In Ordered Vector Spaces

The classical Kantorovich theorem states that every positive operator from a majorizing vector subspace of a Riesz space into a Dedekind complete Riesz space always has a positive extension.We generalize Hahn-Banach and Kantorovich theorem to range space Y being a ordered vector space with Dedekind completeness. On the other hand,using the idea of Abramovich and Wickstead,we give a Kantorovich-type theorem concerning the extension of a positive operator from a normed ordered vector space into a normed ordered vector space with σ-interpolation.Upper and lower semi-continuous functions whose values are in a ordered Banach space are defined.basing on the Michael theorem which is reformulated by Ercan,we present a Hahn-Tong-Kateov theorem concerning the extension of a continuous function from compact Hausdorff space to an ordered Banach space with some additional conditions.It generalize Ercan's and Tietze theorem.We also show that a subset F of C(K, E) is relatively compact whenever F is both equicontinuous and uniformly bounded.And it's converse is false.where K is a compact Hausdorff space,E is a Banach space.If E is a finite-dimension space,then F is relatively compact iff F is both equicontinuous and uniformly bounded.At last,some characterizations of the relatively compactness of subsets in C(K, c_o) and C(K, l_p)(1≤ p < ∞) are deduced.