Some geometric and topological properties of Banach spaces via ball coverings

Abstract

By a ball-covering B of a Banach space X, we mean that B is a collection of open (or closed) balls off the origin whose union contains the unit sphere of X; and X is said to have the ball-covering property provided it admits a ball-covering of countably many balls. This paper shows that G δ property of points in a Banach space X endowed with the ball topology is equivalent to the space X admitting the ball-covering property. Moreover, smoothness, uniform smoothness of X can be characterized by properties of ball-coverings of its finite dimensional subspaces.

Ball covering property from commutative function spaces to non-commutative spaces of operators

“…Bentuo Zheng was supported in part by Simons Foundation Grant 585081. all ε > 0, which further reveals the fact that the BCP is a geometric property deeply related to the weak star topology for dual spaces. The BCP is also widely linked to a number of important properties, such as the G δ property of points in X, Radon-Nikodym property [4], uniform convexity, uniform non-squareness, strict convexity and dentability [13,14], and universal finite representability and B-convexity [16].…”

Characterizations of universal finite representability and b-convexity of Banach spaces via ball coverings

“…We say that X has the ball-covering property if it admits a ball-covering of countably many balls. This notion was introduced by Cheng [1] and intensively studied in [1][2][3][4][5][6][7][8][9][10][11][12][13][14]. It was shown in [1] that every Banach space X with ball-covering has a w * -separable dual.…”

A note on ball-covering property of Banach spaces

“…∞ [1] is a typical example of non-separable space with the BCP. In the recent years the study of the BCP of Banach spaces has attracted mathematicians attention, and the BCP and its applications have been intensively studied in [1][2][3][4][5][6][7]9].…”