On relations between weak approximation properties and their inheritances to subspaces

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⊥ is complemented in the dual space X∗, where M⊥ = {x∗ ∈ X∗: x∗(m) = 0 for all m ∈ 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 M has the weak approximation property (respectively, bounded weak approximation property). © 2006 Elsevier Inc. All rights reserved.