In this note we study the property (V) of Pelczynski, in a Banach space X, in relation with the presence, in the dual Banach spaceX*, of suitableweak*basic sequences. We answer negatively to a question posed by John and we prove that, ifXis a Banach space with the Property (V) of Pelczynski and the Gelfand Phillips property, thenXis reflexive if and only if every quotient with a basis is reflexive. Moreover, we prove that, ifXis a Banach space with the property (V) of Pelczynski, then eitherXis a Grothendieck space orW(X, Y) is uncomplemented inL(X, Y) provided thatYis a Banach space such thatW(X, Y) ≠L(X, Y). [ABSTRACT FROM PUBLISHER]