On the properties (wL) and (wV)

Abstract

Abstract We characterize Banach spaces X with spaces with property (wL), i.e. spaces with the property that every L-subset of X* is weakly precompact. We prove that a Banach space X has property (wL) if and only if for any Banach space Y, any completely continuous operator T : X → Y has weakly precompact adjoint if and only if any completely continuous operator T : X → ℓ ∞ has weakly precompact adjoint. We prove that if E is a Banach space and F is a reflexive subspace of E* such that ⊥ F has property (wL), then E has property (wL). We show that a space E has property RDP* (resp. the DPrcP) if and only if any closed separable subspace of E has property RDP* (resp. the DPrcP). We also show that G has property (wL) if under some conditions K w* (E*, F) contains the dual of G.