On Super Weak Compactness of Subsets and its Equivalences in Banach Spaces

Analogous to weak compactness of subsets of Banach spaces and to property of subsets in super reflexive spaces, the purpose of this paper is to discuss super weak compactness of both convex and nonconvex subsets in Banach spaces. As a result, we give three characterizations of super weakly compact sets: The first one is Grothendiek's type theorem; the second one is James' type characterization and the last one is super Banach-Saks property. We also show that super weak compactness, finite index property and finite dual index property of a closed convex set are actually equivalent. Therefore, eleven notions and properties eventually coincide for a closed bounded convex set. We also present some characterizations of uniformly weakly null sequences. These are done by localizing some basic properties of ultrapowers and using some geometric procedures of Banach spaces.