A dual geometric characterization of Banach spaces not containingl₁

On Localized Weak Precompactness in Banach Spaces

“…In [2] , Bator and Lewis have made a systematic study of localized weak precompactness and obtained its various characterizations which are analogous results of Fitzpatrick [3], Saab [20] and Saab and Saab [21]. Well, in a series of our papers [10], [11], [12], [13] and [14], we have obtained a number of characterizations of Pettis sets with the help of our function in [10].…”

Some characterizations of weak Radon-Nikodým sets

“…The following was shown in [12] about a weak*-compact convex subset A" of the dual E* of a Banach space £: every subset M of A" is a weak*-dentable in (E*, o(E*, £**)) if and only if every x** in E** restricted to any weak*-compact subset M of K has a point of continuity on (M, weak*).…”

“…Recall [12] that a bounded subset M of the dual E* is weak*-dentable in (£*, o(E*, E**)), if for every zero neighborhood V in (E*, o(E*, E**)) there is a weak*-open slice S of M such that S -S C F where S is S = \x* G M; x*(x0) > sup x*(x0) -a\ 1 jc*eM J for ;c0 G £ and a > 0.…”

“…Hence T factors through a Banach space not containing /, [12] and therefore {T(xn); n > 1} is weakly precompact by [10]. This means that there is a subsequence (xn ) such that Txn -xn , converges pointwise on K, that is lim^ x*(xn ) exists for every x* Ei.…”

Representations of dynamical systems on Banach spaces not containing $l_{1}$

“…Then by [37,Theorem 9], A has the scalar point of continuity property if and only if A is a weak Radon-Nikodým subset (WRN for short). We refer to [3,41,37] for exact definitions and additional information about WRN subsets. See also Theorem 6.5 below about WRN compact spaces.…”