Geodesic
Representable Banach Spaces and Uniformly Gateaux-Smooth Norms
Serdica Mathematical Journal, Tome 22 (1996) no. 1, pp. 33-38.

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It is proved that a representable non-separable Banach space does not admit uniformly Gâteaux-smooth norms. This is true in particular for C(K) spaces where K is a separable non-metrizable Rosenthal compact space.
Keywords: Gâteaux-Smoothness, Renormings, Representable Banach Spaces 
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Frontisi, Julien. Representable Banach Spaces and Uniformly Gateaux-Smooth Norms. Serdica Mathematical Journal, Tome 22 (1996) no. 1, pp. 33-38. http://geodesic.mathdoc.fr/item/SMJ2_1996_22_1_a3/