Star-finite coverings of Banach spaces
Functional Analysis
2020-02-12 v1
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 by our results that an LUR or a uniformly Fr\'echet smooth infinite-dimensional Banach space does not admit star-finite coverings by closed balls. On the other hand, we present a quite involved construction proving existence of a star-finite covering of by Fr\'echet 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.
Keywords
Cite
@article{arxiv.2002.04308,
title = {Star-finite coverings of Banach spaces},
author = {Carlo Alberto De Bernardi and Jacopo Somaglia and Libor Vesely},
journal= {arXiv preprint arXiv:2002.04308},
year = {2020}
}