On The Sum Of Proximinal Sets And The Approximative Compactness Of Subspaces In Banach Spaces

First, we investigate the sum of various proximinal sets, including:proximinal sets, strongly proximinal sets, approximatively compact sets and approximatively weakly compact sets, etc. We show in a Banach space that the sum of a weakly compact convex set and a proximinal convex set is proximinal. This generalizes the classical result which says that the sum of a reflexive subspace and a proximinal subspace (satisfying the sum is closed) is again proximinal. We also prove that the sum of a weakly compact convex set and an approximatively weakly compact convex set is also approximatively weakly compact; the sum of a compact set and a strongly proximinal set is strongly proximinal. And, in any infinite-dimensional Banach space, we construct two strongly proximinal sets satisfying that their sum is closed but not proximinal.Secondly, we investigate the approximative compactness of the unit ball of a subspace in Banach spaces. Let Y be an approximatively compact subspace of a Banach space X, and Bx, BY be the closed unit ball of X and Y respectively. We prove that BY is approximatively compact in X if and only if for every translation YT of Y with YT∩Bx≠φ, YT∩Bx is approximatively compact in YT. This is a supplement to the well-known results on the characterizations of the (strongly) ball proximinal hyperplanes. We also obtain some stability results of the approximative compactness of BY (as well as Y) under infinite lp-direct sum, where1≤p≤∞. And as for the approximatively weak compactness, we have similar results as above.