in this paper  we give newresults on the best approximation  in the hausdorff topological vectorspace and  consider relationship  with orthogonality. also we determined under  what conditions the map $p_{k,f}$ is upper semicontinous.

As a counterpart to best approximation, a new kind of approximation, called best coapproximation was introduced in normed linear spaces by C. Franchetti and M. Furi. In this paper, we use this coapproximation to prove some results on the existence and uniqueness of best coapproximation in quotient spaces when the underlying spaces are metric linear spaces. We shall also see how coproximinality ...