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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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