Some Approximation Problems in Abstract Spaces

Approximation Theory is an old and rich branch of analysis. It consists of the theory of the nearest points(best approximation) and the theory of farthest points (worst approximation).Many results are known on the existence, uniqueness and characterization of elements of best approximation and elements of farthest points and related problems when the underlying spaces are normed linear spaces. The consideration of approximation problems in more general spaces (linear metric spaces, topological spaces, metric spaces etc.) is quite challenging and the study although initiated long ago has not reached a satisfactory stage as in the case of normed linear spaces. So the results obtained in more general spaces do not constitute a unified theory and construction of such a theory up to the present is an open problem. The aim of this work is to make a step towards development of this theory in metric spaces, convex metric spaces, M-spaces, externally convex spaces and strongly externally convex spaces.