Hyperconvex metric spaces

Razafindrakoto, Ando Desire (2010-03)

Thesis (MSc (Mathematics))--University of Stellenbosch, 2010.

Thesis

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.

Please refer to this item in SUNScholar by using the following persistent URL: http://hdl.handle.net/10019.1/4106
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