Convergence of best approximations from unbounded sets

Abstract

Given a metric space whose bounded sets are relatively compact (i.e., have compact closures), we show that a nearest point selection from a sequence of Kuratowski converging sets converges to the nearest point in the limit set whenever the latter point is unique. The result is extended to Kuratowski limits of linear varieties in infinite dimensional Hilbert spaces where this nearest point (relative to the origin) is necessarily unique. Finally, we show that the Kuratowski limit of hyperplanes must itself be a hyperplane and that a necessary and sufficient condition for the associated nearest points to the origin to converge as above is that the canonial points parametrizing the hyperplanes converge.