Super weak compactness and uniform Eberlein compacta
type: Item
instance of: scholarly article
title: Super weak compactness and uniform Eberlein compacta
zbMATH Open document ID: 1381.46017
author: Zhi Tao Yang
author: Yufeng Lu
author: Qingjin Cheng
publication date: 2017-08-03T00:00:00Z
review text: A (weakly closed) subset \(K\) of a Banach space \(X\) is said to be super weakly compact (SWC) if for any free ultrafilter \(\mathcal{U}\) on \(\mathbb{N}\), the set \(K^{\mathcal{U}}=\{[x_i]: x_i\in K\text{ for every }i\in\mathbb{N}\}\) is a weakly compact subset of the ultrapower \(X^\mathcal{U}\) of \(X\). A Banach space \(X\) is said to be super weakly compactly generated if there is an SWC convex set \(K\subset X\) with \(\overline{\text{span}} K = X\). The main results of the paper are the following three theorems. {\parindent=0.8cm \begin{itemize}\item[--] Every SWC set is homeomorphic to an SWC set in the space \(c_0(\Gamma)\).\item[--] A compact space is uniformly Eberlein compact (that is, homeomorphic to a weakly compact set in a Hilbert space) if and only if it is homeomorphic to an SWC subset \(K\) of a Banach space such that \(\overline{\text{co}} K\) is SWC\item[--] A Banach space \(X\) is super WCG if and only if there is a Banach space \(Y\) and a continuous linear operator \(T:X^*\to Y\) with \(T|_{B_{X^*}}\) being a \(w^*\)-\(w\) homeomorphism onto an SWC subset of \(Y\). \end{itemize}} The proofs depend on earlier results known in the field and, in the light of those results, are not very difficult.
reviewed by: Marek Cúth
Mathematics Subject Classification ID: 46B26
Mathematics Subject Classification ID: 46B50
zbMATH DE Number: 6756285
zbMATH Keywords: Banach space
zbMATH Keywords: uniform Eberlein compact
zbMATH Keywords: super weak compactness
MaRDI profile type: Publication
full work available at URL: s10114-017-6354-5
OpenAlex ID: W2563904829
cites work: The structure of weakly compact sets in Banach spaces
cites work: On the Structure of Weakly Compact Subsets of Hilbert Spaces and Applications to the Geometry of Banach Spaces
cites work: Opérateurs uniformément convexifiants
cites work: Q4938152
cites work: Continuous images of weakly compact subsets of Banach spaces
cites work: Embedding weakly compact sets into Hilbert space
cites work: Every weakly compact set can be uniformly embedded into a reflexive Banach space
cites work: Q4583049
cites work: On super-weakly compact sets and uniformly convexifiable sets
cites work: On super fixed point property and super weak compactness of convex subsets in Banach spaces
cites work: On Weakly Compact Subsets of Banach Spaces
cites work: Factoring weakly compact operators
cites work: Q3254057
cites work: Banach spaces which can be given an equivalent uniformly convex norm
cites work: On weak compactness in \(L_1\) spaces
cites work: The structure of uniformly Gâteaux smooth Banach spaces
cites work: Sigma-finite dual dentability indices
cites work: Functional analysis and infinite-dimensional geometry
cites work: Q5640213
cites work: Q4136031
cites work: Martingales with values in uniformly convex spaces
cites work: Q3515804
cites work: Super WCG Banach spaces
cites work: Q4052703
DOI: 10.1007/S10114-017-6354-5
published in: Acta Mathematica Sinica, English Series
description: scientific article; zbMATH DE number 6756285