On relations between weak approximation properties and their inheritances to subspaces

Abstract

It is shown that for the separable dual X* of a Banach space X, if X* has the weak approximation property, then X* has the metric weak approximation property. We introduce the properties W*D and MW*D for Banach spaces. Suppose that M is a closed subspace of a Banach space X such that M-L is complemented in the dual space X*, where M-perpendicular to={x*epsilon X*:x*(m)=0 for all m epsilon M}. Then it is shown that if a Banach space X has the weak approximation property and W*D (respectively, metric weak approximation property and MW*D), then At has the weak approximation property (respectively, bounded weak approximation property). (c) 2006 Elsevier Inc. All rights reserved.