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 ⊥ 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).