Pelczynski's property (V) and weak* basic sequences

Abstract

In this note we study the property (V) of Pelczynski, in a Banach space X, in relation with the presence, in the dual Banach space X*, of suitable weak* basic sequences. We answer negatively to a question posed by John and we prove that, if X is a Banach space with the Property (V) of Pelczynski and the Gelfand Phillips property, then X is reflexive if and only if every quotient with a basis is reflexive. Moreover, we prove that, if X is a Banach space with the property (V ) of Pelczynski, then either X is a Grothendieck space or W(X, Y ) is uncomplemented in L(X, Y ) provided that Y is a Banach space such that W(X, Y ) ≠ L(X, Y ).

Keywords: Property (V), Grothendieck space, weak* basic sequences.