Some Geometric Best Approximation and Extension Properties in Banach Spaces

Abstract

This thesis deals with some geometric, best approximation and extension properties newlinein Banach space. In particular, several known geometric properties of a Banach newlinespace are related with notions from best approximation theory. Further, some geometric newlineproperties are introduced and studied. Additionally, extension properties are characterized newlineusing various geometric and best approximation theoretic properties. newlineIn the first part of the thesis, we explore the geometric properties of Banach newlinespaces using best approximation theoretic properties. We present some sufficient and newlineequivalent conditions for a Banach space to be k-uniformly rotund and k-locally uniformly newlinerotund in terms of geometric and best approximation properties. In particular, newlinek-uniform rotundity is characterized in terms of k-uniformly strongly Chebyshevness newlineof the collection of hyperspaces and collection of k-dimensional subspaces. We establish newlinesome relationships of the notions k-locally uniform rotundity and k-midpoint newlinelocally uniform rotundity in terms of k-strongly Chebyshevness. As a consequence, newlinewe obtain an equivalent condition for an and#8467;p-direct sum to be k-midpoint locally uniformly newlinerotund. Further, some characterizations of spaces which are k-uniformly rotund newlinewith respect to a k-dimensional subspace are obtained in terms of k-uniformly strongly newlineChebyshevness and property k-UC.We prove that the notions k-strongly Chebyshevness newlineand k-uniformly strongly Chebyshevness coincide for finite co-dimensional subspaces. newlineSubsequently, we introduce two geometric notions called k-weakly uniform rotundity newlineand k-weakly locally uniform rotundity. These notions are weaker to k-uniform newlinerotundity and k-locally uniform rotundity, also generalizations of the well-known concepts newlineweakly uniform rotundity and weakly locally uniform rotundity in the Sullivan newlinesense. These geometric properties are characterized in terms of suitably defined best newlineapproximation notions namely k-weakly strong Chebyshevity and k-weakly uniform newlinestrong Chebyshevity. We extend some of the existing.