On Stability of the Metric Projection Operator

Abstract. Let M be a closed linear subspace of a normed linear space X. For a given f ∈ X denote by PM f the set of best approximations to f from M . The operator PM is termed the metric projection onto M . In this paper we are interested in the stability of the metric projection PM relative to perturbations of the subspace M . We mainly consider the case where X = Lp, p ∈ [1,∞]. We consider a measure of distance d(M,N) between subspaces M and N and estimate ‖PM f−PNf‖ in terms of d(M,N) and ‖f‖. Typically such an estimate will be of order d(M,N)β with some β which, in general, depends on the geometry of the space X.