Geodesic
Uniform Eberlein Compacta and Uniformly Gâteaux Smooth Norms
Serdica Mathematical Journal, Tome 23 (1997) no. 3-4, pp. 351-362.

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It is shown that the dual unit ball BX∗ of a Banach space X∗ in its weak star topology is a uniform Eberlein compact if and only if X admits a uniformly Gâteaux smooth norm and X is a subspace of a weakly compactly generated space. The bidual unit ball BX∗∗ of a Banach space X∗∗ in its weak star topology is a uniform Eberlein compact if and only if X admits a weakly uniformly rotund norm. In this case X admits a locally uniformly rotund and Fréchet differentiable norm. An Eberlein compact K is a uniform Eberlein compact if and only if C(K) admits a uniformly Gˆateaux differentiable norm.
Keywords: Uniform Eberlein Compacta, Uniform Gâteaux Smooth Norms, Weak Compact Generating 
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Fabian, Marián; Hájek, Petr; Zizler, Václav. Uniform Eberlein Compacta and Uniformly Gâteaux Smooth Norms. Serdica Mathematical Journal, Tome 23 (1997) no. 3-4, pp. 351-362. http://geodesic.mathdoc.fr/item/SMJ2_1997_23_3-4_a10/