PELCZYNSKI'S PROPERTY (V) AND WEAK* BASIC SEQUENCES

In this note we study the property (V) of Pelczynski, in a Banach space X,in relation with the presence, in its dual Banach space, of suitable weak-star basic sequences. We answer negatively to a question posed by John and we provethat, if X is a Banach space with the Property (V) of Pelczynski and the GelfandPhillips property, then X is reflexive if and only if every quotient with a basis isreflexive. Moreover, we prove that, if X is a Banach space with the property (V) ofPelczynski, then either X is a Grothendieck space or W(X;Y) is uncomplementedin L(X;Y) provided that Y is a Banach space such that W(X;Y) is not equal to L(X;Y)