Uniform Eberlein Compacta and Uniformly Gâteaux Smooth Norms

* Supported by grants: AV ĈR 101-95-02, GAĈR 201-94-0069 (Czech Republic) and NSERC 7926 (Canada).

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.