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.
| Original language | English |
|---|---|
| Pages (from-to) | 124384-N/A |
| Journal | Journal of Mathematical Analysis and Applications |
| Volume | 491 |
| DOIs | |
| Publication status | Published - 2020 |
Keywords
- Covering of normed space
- Fréchet smooth body
- Locally uniformly rotund norm
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