Problèmes de renormages. (Renorming problems) (Q1085395)

We show that, for every scattered compact space K such that \(K^{(\omega_ 1)}=\emptyset\), where \(\omega_ 1\) design the first uncountable ordinal, there exists on \({\mathcal C}(K)\) an equivalent norm such that the dual norm is LUR. Moreover, if \(K^{(\omega_ 0)}=\emptyset\), where \(\omega_ 0\) designs the first infinite ordinal, the norm on \({\mathcal C}(K)\) can be chosen itself LUR. We deduce from this result that, if X and Y are Banach spaces such that X' and Y' are WCG, then there exists on \(X\otimes_{\epsilon}Y\) an equivalent LUR norm such that the dual norm is also LUR.