ar X iv : m at h / 92 01 23 0 v 1 [ m at h . FA ] 2 4 Ju l 1 99 1 Banach Spaces with Property ( w ) Denny

In this paper, we give two examples, both of which answer the question in the negative. Both examples are James type spaces considered in [1]. They both possess properties stronger than Property (w). The first example has the property that every operator from the space into the dual is compact. In the second example, both the space and its dual have Property (w). In the last section, we consider if Property (w) passes from a Banach space E to C(K,E). This was also dealt with in [9]. We use standard Banach space terminology as may be found in [6]. For a Banach space E, E ′ denotes its dual, and UE its closed unit ball. If F is also a Banach space, then we let L(E,F ) (respectively K(E,F )) denote the space of all bounded linear operators (respectively all compact operators) from E into F . The norm in l is denoted by ‖ · ‖p. Let us also establish some terminology about sequences. If (ei)