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 c_0(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.
Star-finite coverings of Banach spaces / C.A. De Bernardi, J. Somaglia, L. Vesely. - In: JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS. - ISSN 0022-247X. - 491:2(2020).
|Titolo:||Star-finite coverings of Banach spaces|
SOMAGLIA, JACOPO (Secondo)
VESELY, LIBOR (Ultimo)
|Parole Chiave:||Covering of normed space; Frechet smooth body; Locally uniformly rotund norm;|
|Settore Scientifico Disciplinare:||Settore MAT/05 - Analisi Matematica|
|Progetto:||PIANO DI SOSTEGNO ALLA RICERCA 2015-2017 - LINEA 2 "DOTAZIONE ANNUALE PER ATTIVITA' ISTITUZIONALE"|
|Data di pubblicazione:||2020|
|Digital Object Identifier (DOI):||http://dx.doi.org/10.1016/j.jmaa.2020.124384|
|Appare nelle tipologie:||01 - Articolo su periodico|