Sequential continuity in the ball topology of a Banach space

Abstract

Consider the isometric property (P): the restriction to the unit ball of every bounded linear functional is sequentially continuous in the ball topology. We present in this paper a systematic study of this property, which is a sequential version of the well known ball generated property. A separable Banach space is Asplund if and only if there exists an equivalent norm with property (P). However, in nonseparable spaces this equivalence does not work. Examples of nonseparable Banach spaces with property (P) which are not Asplund are given. Also, we exhibit a nonseparable Asplund space, namely the space of continuous functions on the Kunen compact, admitting no equivalent norm with this property. We characterize reflexive spaces as those satisfying that every equivalent norm has property (P), thus improving a previous characterization involving the ball generated property. Finally, we investigate the relationships between property (P) and Grothendieck spaces.