91Ó°ÊÓ

Let \(C\) be a \(w\) -compact set in a Banach space \(X .\) Show that if \(X^{*}\) is \(w^{*}\) -separable, then \(C\) in its \(w\) -topology is metrizable. In particular, \(C\) is separable. Thus we get an alternative proof that a WCG space \(X\) is separable if \(X^{*}\) is \(w^{*}\) -separable.

Let \(Y\) be a closed subspace of a Banach space \(X\) such that \(X / Y\) is separable. Show that \(X\) is WCG if and only if \(Y\) is WCG.

Prove that for every separable Banach space \(X\) there is a compact \(K \subset S_{X}\) such that \(X=\overline{\operatorname{span}}(K) .\) What about such a weak compact set for WCG spaces?