A stability property for locally uniformly rotund renorming

It is proved that C ( K , E ) (the space of all continuous functions on a Hausdorff compact space K taking values in a Banach space E) admits an equivalent locally uniformly rotund norm if C ( K ) and E do so. Moreover, if the equivalent LUR norms on C ( K ) and E are lower semicontinuous with respect to some weak topologies, the LUR norm on C ( K , E ) can be chosen to be lower semicontinuous with respect to an appropriate weak topology. As a consequence we prove that if X and Y are two Hausdorff compacta and C ( X ) , C ( Y ) admit equivalent (pointwise lower semicontinuous) LUR norms, then so does C ( X × Y ) .