Thesis (MSc (Mathematics))--University of Stellenbosch, 2010. / ENGLISH ABSTRACT: One of the early results that we encounter in Analysis is that every metric space admits
a completion, that is a complete metric space in which it can be densely embedded. We
present in this work a new construction which appears to be more general and yet has
nice properties. These spaces subsequently called hyperconvex spaces allow one to extend
nonexpansive mappings, that is mappings that do not increase distances, disregarding the
properties of the spaces in which they are defined. In particular, theorems of Hahn-Banach
type can be deduced for normed spaces and some subsidiary results such as fixed point
theorems can be observed. Our main purpose is to look at the structures of this new type
of “completion”. We will see in particular that the class of hyperconvex spaces is as large
as that of complete metric spaces. / AFRIKAANSE OPSOMMING: Een van die eerste resultate wat in die Analise teegekom word is dat enige metriese ruimte
’n vervollediging het, oftewel dat daar ’n volledige metriese ruimte bestaan waarin die
betrokke metriese ruimte dig bevat word. In hierdie werkstuk beskryf ons sogenaamde
hiperkonvekse ruimtes. Dit gee ’n konstruksie wat blyk om meer algemeen te wees,
maar steeds gunstige eienskappe het. Hiermee kan nie-uitbreidende, oftewel afbeeldings
wat nie afstande rek nie, uitgebrei word sodanig dat die eienskappe van die ruimte
waarop dit gedefinieer is nie ’n rol speel nie. In die besonder kan stellings van die Hahn-
Banach-tipe afgelei word vir genormeerde ruimtes en sekere addisionele ressultate ondere
vastepuntstellings kan bewys word. Ons hoofdoel is om hiperkonvekse ruimtes te ondersoek.
In die besonder toon ons aan dat die klas van alle hiperkonvekse ruimtes net so groot
soos die klas van alle metriese ruimtes is.
| Identifer | oai:union.ndltd.org:netd.ac.za/oai:union.ndltd.org:sun/oai:scholar.sun.ac.za:10019.1/4106 |
| Date | 03 1900 |
| Creators | Razafindrakoto, Ando Desire |
| Contributors | Holgate, David, Kunzi, Hans-Peter A., University of Stellenbosch. Faculty of Science. Dept. of Mathematical Sciences. |
| Publisher | Stellenbosch : University of Stellenbosch |
| Source Sets | South African National ETD Portal |
| Language | English |
| Detected Language | Unknown |
| Type | Thesis |
| Format | 90 p. |
| Rights | University of Stellenbosch |
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