Some converses of the strong separation theorem

Abstract:

A convex subset $B$ of a real locally convex space $X$ is said to have the separation property if it can be separated from every closed convex subset $A$ of $X$, which is disjoint from $B$, by a closed hyperplane. The strong separation theorem says that if $B$ is weakly compact, then it has the separation property. In this paper, we present two versions of the converse and discuss an application of them. For example, we prove that a normed space is reflexive if and only if its closed unit ball has the separation property. Results in this paper can be considered as supplements of the famous theorem of James.