Sur les cones (faiblement complets) contenus dans le dual d'un espace de Banach non-reflexif

Abstract

Let B be a Banach space. consider the convex proper weakly complete cones X contained in B′ with σ(B′, B) such that X ∩ B′, is conic in the sense of Asimow: that is, there exists α ≥ 0 and f ∈ B″ such that ‖ ‖B ≤ f ≤ α·‖ ‖B on X. This class arises in the theory of integral representations.If B is reflexive, such a cone has a weakly-compact basis. This paper considers the converse problem:- if one requires that X ∩ B′1 be σ(B′, B) metrisable, the existence of X (without a compact σ(B′, B) basis) is equivalent to the statement that B is not a Grothendieck space.However, in every space C(K) with infinitely compact K, one can find such a cone X. If two such cones in B′ are not too far apart, their sum belongs to this class.