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Covering spheres of Banach spaces by balls. (English) Zbl 1179.46015
A countable ball-covering in a Banach space \(X\) a is countable collection of balls off the origin such that the union of these balls contains the unit sphere of \(X\). This concept was introduced by L.-X. Cheng in [Isr. J. Math. 156, 111–123 (2006; Zbl 1139.46016)]. It easily follows from the separation theorem that, if \(X\) admits a countable ball-covering, then \(X^*\) is weak*-separable. The converse statement is false: \(\ell_\infty\) in a suitable renorming constructed by L.-X. Cheng, Q.-J. Cheng, and X.-Y. Liu [Sci. China, Ser. A 51, No. 1, 143–147 (2008; Zbl 1152.46010)] is a counterexample.
The authors prove a kind of converse theorem under renorming: if \(X^*\) is weak*-separable, then, for every \(\varepsilon>0\), \(X\) possesses an \((1+\varepsilon)\)-equivalent norm in which \(X\) admits a countable ball-covering. Moreover, the covering in this theorem is formed by closed balls of a fixed radius.
The construction is based on the following proposition. Let \(X\) be an infinite-dimensional Banach space. Then, for every \(\varepsilon > 0\), there is a biorthogonal sequence \(\{x_n, f_n\}_{n \in \mathbb N} \subset X \times X^*\) such that \(w^*\)-\(\lim f_n=0\), \(\|f_n\|=1\), \(\|x_n\|\leq 1 + \varepsilon\).

MSC:
46B20 Geometry and structure of normed linear spaces
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