We study star-finite coverings of infinite-dimensional normed spaces. A family of sets is called star-finite if each of its members intersects only finitely many other members of the family. It follows from our results that an LUR or a uniformly Fréchet smooth infinite-dimensional Banach space does not admit star-finite coverings by closed balls. On the other hand, we present a quite involved construction of a star-finite covering of c0(Γ) by Fréchet smooth centrally symmetric bounded convex bodies. A similar but simpler construction shows that every normed space of countable dimension (and hence incomplete) has a star-finite covering by closed balls.
De Bernardi, C. A., Somaglia, J., Vesely, L., Star-finite coverings of Banach spaces, <<JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS>>, 2020; 491 (2): 124384-N/A. [doi:10.1016/j.jmaa.2020.124384] [http://hdl.handle.net/10807/164334]
Star-finite coverings of Banach spaces
De Bernardi, Carlo Alberto;
2020
Abstract
We study star-finite coverings of infinite-dimensional normed spaces. A family of sets is called star-finite if each of its members intersects only finitely many other members of the family. It follows from our results that an LUR or a uniformly Fréchet smooth infinite-dimensional Banach space does not admit star-finite coverings by closed balls. On the other hand, we present a quite involved construction of a star-finite covering of c0(Γ) by Fréchet smooth centrally symmetric bounded convex bodies. A similar but simpler construction shows that every normed space of countable dimension (and hence incomplete) has a star-finite covering by closed balls.
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