Abstract
If the unit sphere of a Banach space X can be covered by countably many balls no one of which contains the origin, then, as an easy consequence of the separation theorem, X* is w*-separable. We prove the converse under suitable renorming. Moreover, the balls of the countable covering can be chosen as translates of the same ball.
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Research of V. P. Fonf was supported in part by Israel Science Foundation, Grant # 139/02 and by the Istituto Nazionale di Alta Matematica of Italy. Research of C. Zanco was supported in part by the Ministero dell’Università e della Ricerca Scientifica e Tecnologica of Italy and by the Center for Advanced Studies in Mathematics at the Ben-Gurion University of the Negev, Beer-Sheva, Israel.
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Fonf, V.P., Zanco, C. Covering spheres of Banach spaces by balls. Math. Ann. 344, 939–945 (2009). https://doi.org/10.1007/s00208-009-0336-6
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DOI: https://doi.org/10.1007/s00208-009-0336-6