Erratum to: Ball-covering property of Banach spaces

Abstract

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 BX∗ of a dual space X ∗ under the assumption that X∗ is w∗-separable and use w∗-sequential compactness of BX∗ with BX∗ being w∗-angelic. However, as pointed out by M. Fabian, w∗-separability of X∗ does not imply w∗-separability of BX∗ [6] (see, also [8]), and w ∗-separability of BX∗ does not entail that BX∗ is w∗-angelic in general. Recently, V. Kadets also informed me in an email that the closed unit ball of ∞ endowed with the equivalent norm, defined as the mean of the natural norm of ∞ and the quotient norm of /c0 , is such a counterexample (see his review of [4] in Zentralblatt Math.: Zbl 1152.46010). Recall a collection B of open (closed) balls in a Banach space X is said to be a ball-covering of X if every ball in B does not contain the origin and B covers the unit sphere SX of X . We say that X has the ball-covering property, if it admits a ball-covering of countably many balls. Some authors consider ball-coverings by open balls, the others — by closed ones. Since every open ball is a countable union of closed balls, and every closed ball off the origin is contained in an open ball off the origin with almost the same radius, we have the following simple property, which says that it does not matter much which of the two definitions one uses. ∗ Support by the Natural Science Foundation of China, grant 10771175. Received September 4, 2009 and in revised form February 19, 2010