On the Krein-Milman theorem in vector spaces over a non-Archimedean valued field \(K\) (Q1813429)

Assume that \(K\) is a nonarchimedian complete valued field with a nontrivial valuation, and \(E\) is a Hausdorff locally convex space over \(K\). In this paper some Krein-Milman like theorems for sets in \(E\) are proved. The considered sets are absolutely convex, closed and weakly \(c'\)-compact, if \(K\) is spherically complete, and absolutely convex, closed and \(c'\)-compact, if \(K\) is not spherically complete. For such a set \(A\subseteq E\), the minimal elements of the set \(B=\{S\subseteq A\mid\;\overline{co}(S)=A\}\) are selected.