Super weak compactness and uniform Eberlein compacta
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.
scientific article; zbMATH DE number 6756285
scientific article