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A counterexample to several questions about scattered compact spaces. (English) Zbl 0725.46007
The author constructs a compact scattered space K with the following “negative” properties:
1) K does not have the Namioka property,
2) the Banach space C(K) admits no Gateaux differentiable renorming,
3) C(K) admits no strictly convex renormings.
Let us mention that a compact space K is said to have the Namioka property if, for every Baire space B and every separately continuous function $$\phi$$ : $$B\times K\to {\mathbb{R}}$$, there is a dense $$G_{\delta}$$ subset H of B such that $$\phi$$ is (jointly) continuous at all points of $$H\times K$$.

MSC:
 46B03 Isomorphic theory (including renorming) of Banach spaces 54D30 Compactness 46B20 Geometry and structure of normed linear spaces
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