Semi-continuity of metric projections in ℓ_{∞}-direct sums

Abstract

Let Y Y be a proximinal subspace of finite codimension of c 0 c_0 . We show that Y Y is proximinal in ℓ ∞ \ell _\infty and the metric projection from ℓ ∞ \ell _\infty onto Y Y is Hausdorff metric continuous. In particular, this implies that the metric projection from ℓ ∞ \ell _\infty onto Y Y is both lower Hausdorff semi-continuous and upper Hausdorff semi-continuous.