Institute of Mathematics and Informatics Bulgarian Academy of Sciences
Serdica Mathematical Journal, Vol. 23, No 3-4, (1997), 351p-362p
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.
* Supported by grants: AV ĈR 101-95-02, GAĈR 201-94-0069 (Czech Republic) and NSERC 7926 (Canada).