On James boundaries in dual Banach spaces

Positively Norming Sets in Banach Function Spaces

“…The size of norming sets for Banach spaces has shown to be directly related to the L 1 -structure of their subspaces. The reader can find in the references [5,14,15,16] a deep analysis regarding the existence of isomorphic copies of 1 in Banach spaces with small norming sets. In the case of positively norming sets, this relation is even more direct.…”

“…These examples and the philosophy of the results in [5], [14], [15], [16] might lead to the thinking that if for some positively norming set N , co w * (N ) is small compared to B X * + then X must contain 1 . The following example shows that this is simply not the case:…”

“…On the one hand, there are several recent results relating the weak* closure of norming sets with embedding isomorphic copies of 1 ( [5,15]). In this direction, in this paper we will show that in fact the 1 -structure of a Banach lattice is closely related to the compactness of its positively norming sets (see Theorem 3.2).…”