The aim of our present note is to show the strength of the existence of an
equivalent analytic renorming of a Banach space, even compared to C∞-Fréchet smooth
renormings.
It was Haydon who first showed in [8] that C(K) spaces for K countable admit
an equivalent C∞-Fréchet smooth norm. Later, in [7] and [9] he introduced a large
clams of tree-like (uncountable) compacts K for which C(K) admits an equivalent
C∞-Fréchet smooth norm.
Recently, it was shown in [3] that C(K) spaces for K countable admit...
* 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....
It is proved that no convex and Fréchet differentiable function on c(w), whose derivative is locally uniformly continuous, attains its minimum at a unique point.
We prove that a Banach space admitting an equivalent WUR norm is an Asplund space. Some related dual renormings are also presented.
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