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Ball-covering property of Banach spaces. (English) Zbl 1139.46016
The paper deals with coverings of the unit sphere of a Banach space $$X$$ by a collection of balls, not containing the origin. The main results are: (1) if $$\text{dim} X = n < \infty$$, then the number of balls in such a covering cannot be smaller than $$n+1$$, and if, moreover, $$X$$ is smooth, then there is such a covering by exactly $$n+1$$ balls. (2) If $$S_X$$ admits such a covering by a countable number of balls $$B(x_n,r_n),$$ $$n \in \mathbb N$$, then $$X^*$$ is weak-star separable, and if additionally $$\sup_n r_n< 1$$, then $$X$$ is separable. (3) The unit sphere of $$\ell_\infty$$ can be covered by a countable collection of radius-one balls not containing the origin. Finally, (4) if $$X$$ is a Gâteaux differentiability space with weak-star separable $$X^*$$, then $$S_X$$ admits such a covering by a countable number of balls (Theorem 4.3 of the paper).
Also, in several places (Lemma 4.2, Theorem 4.3, Corollary 4.4, Theorem 4.5 and Theorem 4.7) the author confuses weak-star separability of $$X^*$$ with weak-star separability of $$B_{X^*}$$. Also there are some unclear places in the proof of Lemma 4.2, where the author confuses weak-star sequential compactness of $$B_{X^*}$$ with the condition that for every $$A \subset B_{X^*}$$ every weak-star cluster point of $$A$$ is the weak-star limit of a sequence $$(a_n) \subset A$$ (i.e., that $$(B_{X^*}, w^*)$$ is angelic). This obstacle renders the validity of Theorem 4.3 questionable.

##### MSC:
 46B20 Geometry and structure of normed linear spaces
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##### References:
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