Star-finite coverings of Banach spaces

Carlo Alberto De Bernardi; Jacopo Somaglia; Libor Veselý

doi:10.1016/j.jmaa.2020.124384

Star-finite coverings of Banach spaces

Carlo Alberto De Bernardi
, Jacopo Somaglia
, Libor Veselý

University of Milan

Risultato della ricerca: Contributo in rivista › Articolo

1 Citazioni (Scopus)

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.

Lingua originale	Inglese
pagine (da-a)	124384-N/A
Rivista	Journal of Mathematical Analysis and Applications
Volume	491
DOI	https://doi.org/10.1016/j.jmaa.2020.124384
Stato di pubblicazione	Pubblicato - 2020

Keywords

Covering of normed space
Fréchet smooth body
Locally uniformly rotund norm

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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 Fr{\'e}chet 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 Fr{\'e}chet 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 = "Covering of normed space, Fr{\'e}chet smooth body, Locally uniformly rotund norm, Covering of normed space, Fr{\'e}chet smooth body, Locally uniformly rotund norm",

author = "\{De Bernardi\}, \{Carlo Alberto\} and Jacopo Somaglia and Libor Vesel{\'y}",

year = "2020",

doi = "10.1016/j.jmaa.2020.124384",

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T1 - Star-finite coverings of Banach spaces

AU - De Bernardi, Carlo Alberto

AU - Somaglia, Jacopo

AU - Veselý, Libor

PY - 2020

Y1 - 2020

N2 - 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.

AB - 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.

KW - Covering of normed space

KW - Fréchet smooth body

KW - Locally uniformly rotund norm

KW - Covering of normed space

KW - Fréchet smooth body

KW - Locally uniformly rotund norm

UR - http://hdl.handle.net/10807/164334

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DO - 10.1016/j.jmaa.2020.124384

M3 - Article

SN - 0022-247X

VL - 491

SP - 124384-N/A

JO - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

ER -

Star-finite coverings of Banach spaces

Abstract

Keywords

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