Helly's selection theorem and the principle of local reflexivity of ordered type

Abstract

Let ( E, E +,∥ · ∥) be an ordered normed space with a positive cone E +, let 0 ≤ ψ ϵ E″, let N be finite-dimensional subspace of E′ and ε > 0. In terms of the notions of half-full injections and half-decomposable surjections, sufficient conditions for N to ensure the existence of x ϵ E + with ∥ x∥≤∥ψ∥ + ϵ and ψ= K E x on N have been found (Theorems 3.5 and 3.6). As an application of Helly's selection theorem of ordered type, the principle of local reflexivity of ordered type is obtained (Theorem 4.7).