A note on lower semicontinuous smooth norms

In Fabian et al. (Arch Math (Basel) 78(6):459–464, 2002) it was proved that, if a Valdivia compact space K has property \(\mathcal P\) —a property defined in terms of the Gâteaux differentiability of certain norms on C(K)—then, under a cardinality condition on the density character, C(K) is a weakly Lindelof determined space, and in particular, K is Corson. We prove here that the result holds in general without cardinality restrictions. Moreover, the proof given here is much simpler than the original one.