Every Banach space with a w *-separable dual has a 1+ɛ-equivalent norm with the ball covering property

Abstract

A normed space is said to have ball-covering property if its unit sphere can be contained in the union of countably many open balls off the origin. This paper shows that for every ε > 0 every Banach space with a w * -separable dual has a 1+ε-equivalent norm with the ball covering property.Citation: Cheng L X, Shi H H, Zhang W. Every Banach space with a w * -separable dual has a 1 + ε-equivalent norm with the ball covering property.

Erratum to: Ball-covering property of Banach spaces

“…Sufficiency is trivial by the proposition above and by Theorem 2.6 of [2] (see also [3]). Necessity.…”

“…Though several conclusions of [1] (i.e., Lemma 4.2, Theorem 4.3, Corollary 4.4, Theorem 4.5 and Theorem 4.7) are true in a renorming sense using a recent theorem either by V. Fonf and C. Zanco [5] or by L. Cheng, H. Shi and W. Zhang [3], the proofs of the mentioned results in [1] use w * -separability of the unit ball B X * of a dual space X * under the assumption that X * is w * -separable and use w * -sequential compactness of B X * with B X * being w * -angelic. However, as pointed out by M. Fabian, w * -separability of X * does not imply w * -separability of B X * [6] (see, also [8]), and w * -separability of B X * does not entail that B X * is w * -angelic in general.…”

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.…”

“…Conversely there exists a Banach space with a w * -separable dual which does not admit ball-covering property in general. Recently, Cheng and Shi [2], Fonf and Zanco [3] showed independently that every Banach space X with a w * -separable dual is always approximated uniformly by norms with ball-covering property. Article [4] showed a characterization of uniformly non-square Banach spaces by ball-coverings.…”

Some geometric and topological properties 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. In the recent years, the study of the ball-covering property of Banach spaces has also brought mathematicians attention, and such property has been intensively studied in [3][4][5][6][7][8]11,[13][14][15][16]20,21].…”