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

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