A Banach space admits a locally uniformly rotund norm if its dual is a Vašák space (Q1813673)

A Banach space \(X\) is said to be weakly countably determined (or a Vašák space) if there is a countable family \(\{K_ n\}\) of \(w^*\)-compact subsets of \(X^{**}\) such that for every \(x\in X\) and every \(z\in X^{**}\setminus X\), there is \(n_ 0\) such that \(x\in K_{n_ 0}\) and \(z\not\in K_{n_ 0}\). It is proved that \(X\) admits an equivalent locally uniformly rotund norm whenever \(X^*\) is weakly countably determined.