Weakly metrizability of spheres and renormings of Banach spaces (Q2794436)

Let \(X\) be a (rather nonseparable) Banach space, \(F\subset X^*\) a subspace of the topological dual of \(X\) which is norming (that is, there exists \(r>0\) with \(\|x\|\leq r\sup\{|x^*(x)|:\; x^*\in F, \,\|x^*\|\leq 1\}\) for every \(x\in X\)) and denote by \(\sigma(X,F)\) the topology on \(X\) of pointwise convergence on \(F\).NEWLINENEWLINEThe authors study the metrizability of \((S_X,\sigma(X,F))\). They prove three interesting and deep theorems which are too involved to be stated here in a detail. Let us roughly decribe the obtained results. NEWLINE{\parindent=7mmNEWLINE\begin{itemize}\item[1.] An equivalent condition to ``the existence of a \(\sigma(X,F)\)-lsc equivalent norm such that \((S_X,\sigma(X,F))\) is metrizable'' is given. NEWLINE\item[2.] An equivalent condition to ``(i) + basis of \((S_X,\sigma(X,F))\) is made of certain slices'' is given. NEWLINE\item[3.] Four equivalent conditions to ``\(X^*\) admits an equivalent dual norm such that \((S_{X^*},w^*)\) is metrizable'' are found.NEWLINENEWLINE\end{itemize}} NEWLINEIn particular, \((B_{X^*},w^*)\) is descriptively compact if and only if \(X^*\) admits an equivalent dual norm such that \((S_{X^*},w^*)\) is metrizable.