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

Nthebe, Johannes M. T. (2006-12)

Thesis (MSc) -- University of Stellenbosch, 2006.

Thesis

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-continuous measure v on ∑; with bounded variation │v│, there exists a Bochner integrable function f : 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 is den table. Futhermore, a Banach space X has the Krein-Milman property (KMP) if each closed bounded 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 Banach space 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. In this thesis we examine conditions under which the KMP implies the RNP.

AFRIKAANSE OPSOMMING: Gestel X is 'n Banach-ruimte oor die liggaam R Dan het X die Radon-Nikodym-eienskap as vir elke eindige positive maatruimte (0, E, µ) en vir elke µ-kontinue aftelbaar additiewe maat v: E ~ X met begrensde variasie lvl, daar 'n Bochner-integreerbare funksie f: 0 ~ X bestaan sodanig dat v(E) = fe fdµ vir elke EE E. Die RN-eienskap het van die maatteoretiese na die meetkundige verskuif toe Rieffel aangetoon het dat 'n Banach-ruimte X die Radon-Nikodym eienskap het as en slegs as elke nie-lee begrensde deelversameling van X induikbaar is. Verder, 'n Banach-ruimte X het die Krein-Milman-eienskap as elke nie-lee geslote begrensde konvekse deelversameling van X gelyk is aan die geslote konvekse omhulsel van sy ekstreempunte. Lindenstrauss het bewys dat as elke nie-lee geslote begrensde konvekse deelversameling van 'n Banach-ruimte X 'n ekstreempunt bevat, dan het X die KreinMilman-eienskap. In die besonder geld dat 'n Banach-ruimte met die Radon-Nikodym eienskap ook die Krein-Milman-eienskap het. Omdat die omgekeerde in die algemeen nie geld nie, word in hierdie tesis onclersoek ingestel na voorwaardes waaronder Krein-Milman wel vir Radon-Nikodym impliseer.

Please refer to this item in SUNScholar by using the following persistent URL: http://hdl.handle.net/10019.1/50668

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