A characterization of the kernel of a closed set

Abstract:

Let $S$ be a closed subset of some linear topological space such that int ker $S \ne \phi$ and ker $S \ne S$ Let $\mathcal {C}$ denote the collection of all maximal convex subsets of $S$ and, for any fixed $k \geq 1$, let $\mathfrak {M} = \{ {A_1} \cup \cdots \cup {A_k}:{A_1}, \ldots ,{A_k}$ distinct members of $\mathcal {C}\}$. Then $\mathfrak {M} \ne \phi$ and $\cap \mathfrak {M} = \ker S$.