Journal of Convex Analysis 25 (2018), No. 3, 899--926
Copyright Heldermann Verlag 2018
On Super Weak Compactness of Subsets and its Equivalences in Banach Spaces
Lixin Cheng
School of Mathematical Sciences, Xiamen University, Xiamen 361005, P.R.China
lxcheng@xmu.edu.cn
Qingjin Cheng
School of Mathematical Sciences, Xiamen University, Xiamen 361005, P.R.China
qjcheng@xmu.edu.cn
Sijie Luo
School of Mathematical Sciences, Xiamen University, Xiamen 361005, P.R.China
Kun Tu
School of Mathematical Sciences, Yangzhou University, Yangzhou 225002, P.R.China
Jichao Zhang
School of Science, Hubei University of Technology, Wuhan 430068, P.R.China
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.
Keywords: Super weakly compact set, super Banach-Saks set, finite index, finite dual index, ultraproduct, Banach space.
MSC: 46B20, 46B03, 46B50
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