A new approach to the Krein-Milman theorem
Some generalizations of the theorem of Krein-Milman were developed in the seventies in order to include other type of topological vector spaces apart from the usual ones (e.g. Kalton's papers within the context of locally p-convex spaces). However, the search of a Krein-Milman theorem has been one of the classical ''open problems'' in non-archimedean analysis [see \textit{A. F. Monna}, Bull. Soc. Math. France, Suppl., Mém. No.39-40, 255-278 (1974; Zbl 0292.46001)]. In fact, it is not difficult to verify that the usual definitions of extreme points are not suitable for the ultrametric case (e.g. the boundary of every open ball in a normed space is empty!). In order to give a theorem of Krein-Milman which includes the case of locally p-convex spaces (\(p\in(0,1])\) and the non-archimedean case, we propose in this paper a new definition of extreme points. The latter definition agrees with the usual one in case the ground field is R or C and it allows us to give a non-archimedean Krein-Milman theorem.
scientific article; zbMATH DE number 3855787