Full Text Available
Note: Clicking the button above will open the full text document at the original institutional repository in a new window.
The Radon-Nikodym and the Krein-Milman properties in Banach spaces
Thesis (MSc) -- University of Stellenbosch, 2006.
Saved in:
| Main Author: | |
|---|---|
| Other Authors: | |
| Format: | Thesis |
| Language: | en_ZA |
| Published: |
Stellenbosch : Stellenbosch University
2012
|
| Subjects: | |
| Tags: |
No Tags, Be the first to tag this record!
|
| _version_ | 1867613737628729344 |
|---|---|
| access_status_str | Open Access |
| author | Nthebe, Johannes M. T. |
| author2 | Maritz, P. |
| author_browse | Maritz, P. Nthebe, Johannes M. T. |
| author_facet | Maritz, P. Nthebe, Johannes M. T. |
| author_sort | Nthebe, Johannes M. T. |
| collection | Thesis |
| dc_rights_str_mv | Stellenbosch University |
| description | Thesis (MSc) -- University of Stellenbosch, 2006. |
| format | Thesis |
| id | oai:scholar.sun.ac.za:10019.1/50668 |
| institution | Stellenbosch University (South Africa) |
| language | en_ZA |
| last_indexed | 2026-06-10T12:40:54.381Z |
| license_str | Other — see source repository |
| provenance_str_mv | Harvested via OAI-PMH from SUNScholar — Stellenbosch University Repository |
| publishDate | 2012 |
| publishDateRange | 2012 |
| publishDateSort | 2012 |
| publisher | Stellenbosch : Stellenbosch University |
| publisherStr | Stellenbosch : Stellenbosch University |
| record_format | dspace |
| source_str | SUNScholar — Stellenbosch University Repository |
| spelling | oai:scholar.sun.ac.za:10019.1/50668 The Radon-Nikodym and the Krein-Milman properties in Banach spaces Nthebe, Johannes M. T. Maritz, P. Stellenbosch University. Faculty of Science. Dept. of Mathematical Sciences. Banach spaces -- Radon-Nikodym property Banach spaces -- Krein-Milman property Dissertations -- Mathematics Thesis (MSc) -- University of Stellenbosch, 2006. 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. 2012-08-27T11:33:32Z 2012-08-27T11:33:32Z 2006-12 Thesis http://hdl.handle.net/10019.1/50668 en_ZA Stellenbosch University 117 pages application/pdf Stellenbosch : Stellenbosch University |
| spellingShingle | Banach spaces -- Radon-Nikodym property Banach spaces -- Krein-Milman property Dissertations -- Mathematics Nthebe, Johannes M. T. The Radon-Nikodym and the Krein-Milman properties in Banach spaces |
| title | The Radon-Nikodym and the Krein-Milman properties in Banach spaces |
| title_full | The Radon-Nikodym and the Krein-Milman properties in Banach spaces |
| title_fullStr | The Radon-Nikodym and the Krein-Milman properties in Banach spaces |
| title_full_unstemmed | The Radon-Nikodym and the Krein-Milman properties in Banach spaces |
| title_short | The Radon-Nikodym and the Krein-Milman properties in Banach spaces |
| title_sort | radon nikodym and the krein milman properties in banach spaces |
| topic | Banach spaces -- Radon-Nikodym property Banach spaces -- Krein-Milman property Dissertations -- Mathematics |
| url | http://hdl.handle.net/10019.1/50668 |
| work_keys_str_mv | AT nthebejohannesmt theradonnikodymandthekreinmilmanpropertiesinbanachspaces AT nthebejohannesmt radonnikodymandthekreinmilmanpropertiesinbanachspaces |