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).