In any dual space X * , the set QP of quasi-polyhedral points is contained in the set SSD of points of strong subdifferentiability of the norm which is itself contained in the set NA of norm attaining functionals. We show that NA and SSD coincide if and only if every proximinal hyperplane of X is strongly proximinal, and that if QP and NA coincide then every finite codimensional proximinal subspace of X is strongly proximinal. Natural examples and applications are provided.

We say that a normed linear space X is a R(1) space if the following holds: If Y is a closed subspace of finite codimension in X and every hyperplane containing Y is proximinal in X then Y is proximinal in X. In this paper we show that any closed subspace of c 0 is a R(1) space. Academic PressSometimes, but not always, finite codimensional subspaces Y of a Banach space X are proximinal when every hyperplane of X containing Y is itself proximinal (see [2]). When this happens, we say, following [5], that X is a R(1) space. Non trivial examples of R(1) spaces are usually obtained through some smoothness condition on the space X* (see e.g.[5], Prop. 1). However, the space c 0 is easily seen to be a R(1) space although its dual l 1 is very far from smooth. We show in this paper that the R(1) property extends to subspaces of c 0 . We will do so by using a remote form of smoothness in l 1 , namely a strong form of subdifferentiabilty for norm attaining functionals in l 1 .We consider only real normed linear spaces. Let X be a normed linear space. Then X* denotes its dual. The closed unit ball and the unit sphere of X are denoted by B(X) and S(X) respectively. By NA(X), we denote the subset of X*, consisting of all the norm attaining functionals on X. For x # X, we setIf Y is a subspace of X, we say that Y is proximinal in X if every x # X has a nearest element from Y. That is, there exists z # Y such that &x&z&=d(x, Y)=inf[&x& y& : y # Y].

No abstract

Abstract. Let X be a closed subspace of c 0 . We show that the metric projection onto any proximinal subspace of finite codimension in X is Hausdorff metric continuous, which, in particular, implies that it is both lower and upper Hausdorff semicontinuous.