Continuity properties of projection operators

We prove that the projection operator on a nonempty closed convex subset[inline-graphic not available: see fulltext] of a uniformly convex Banach spaces is uniformly continuous on bounded sets and we provide an estimate of its modulus of uniform continuity. We derive this result from a study of the dependence of the projection on[inline-graphic not available: see fulltext] of a given point when[inline-graphic not available: see fulltext] varies.