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.
1999 Academic Press
Sometimes, 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 set
If 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].