The Radon-Nikodym and the Krein-Milman properties in Banach spaces

[摘要] ENGLISH ABSTRACT: A Banach space X over the field of real numbers R. has the Radon-Nikodym property(RNP) if for each finite positive measure space (Ω,∑,µ) and each X-valued, µ-continuousmeasure v on ∑; with bounded variation │v│, there exists a Bochner integrable functionf : Ω -- X such that v (E) = ∫ e f dµ for E =∑.The RNP has become a geometrical property when the following result was introduced:A Banach space X has the RNP if and only if each non-empty bounded subset of X isden table.Futhermore, a Banach space X has the Krein-Milman property (KMP) if each closedbounded convex subset of X is the closed convex hull of its extreme points.Lindenstrauss proved that if each nonempty closed bounded convex subset of a Banachspace X contains an extreme point, then X has the Krein-Milman property. In particular,a Banach space with the RNP has the KMP. The converse remains an open question. Inthis thesis we examine conditions under which the KMP implies the RNP.

[发布机构] Stellenbosch University