On stability of the metric projection operator

摘要

Let M be a closed linear subspace of a normed linear space X. For a given f ? X denote by PMf 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 PMf ?PNf in terms of d(M,N) and f. Typically such an estimate will be of order d(M,N)? with some ? which, in general, depends on the geometry of the space X.