A new approach to the Krein-Milman theorem
type: Item
instance of: scholarly article
title: A new approach to the Krein-Milman theorem
zbMATH Open document ID: 0538.46057
author: Javier Martinez Maurica
author: Cristina Pérez-García
published in: Pacific Journal of Mathematics
publication date: 1985-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 46S10
Mathematics Subject Classification ID: 46A55
Mathematics Subject Classification ID: 12J25
zbMATH DE Number: 3855787
zbMATH Keywords: locally p-convex spaces
zbMATH Keywords: Krein-Milman theorem
zbMATH Keywords: non-archimedean case
zbMATH Keywords: extreme points
MaRDI profile type: Publication
full work available at URL: pjm.1985.120.417
OpenAlex ID: W1490295525
DOI: 10.2140/PJM.1985.120.417
description: scientific article; zbMATH DE number 3855787