Proximinality and co-proximinality in metric linear spaces

As a counterpart to best approximation, the concept of best coapproximation was introduced in normed linear spaces by C. Franchetti and M. Furi in 1972. Subsequently, this study was taken up by many researchers. In this paper, we discuss some results on the existence and uniqueness of best approximation and best coapproximation when the underlying spaces are metric linear spaces. A new kind of approximation, called best coapproximation was introduced in normed linear spaces by C. Franchetti and M. Furi [2] to obtain some characterizations of real Hilbert spaces among real Banach spaces. This study was further taken up by many researchers in normed linear spaces and Hilbert spaces (see e.g. [3], [4], [9]). But only a few have taken up this study in more general abstract spaces. The theory of best coapproximation is much less developed as compared to the theory of best approximation in abstract spaces. The present paper is also a step in this direction. In this paper, we discuss the existence and uniqueness results on best approximation and best coapproximation in metric linear spaces thereby generalizing the various known results. We start with a few definitions. 1The research work of the author has been supported by U.G.C., India under Emeritus Fellowship. 2The research work of the author has been supported by U.G.C., India under Senior Research Fellowship. 2010 Mathematics Subject Classification. 41A50, 41A52, 41A65, 35E10.