TY - JOUR
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
U2 - 10.1016/j.jmaa.2020.124384
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 -