Some geometric and topological properties of Banach spaces via ball coverings
The \textit{ball topology} on a Banach space \(X\) was studied by \textit{G. Godefroy} and \textit{N. J. Kalton} [Banach space theory, Proc. Res. Workshop, Iowa City/Iowa 1987, Contemp. Math. 85, 195--237 (1989; Zbl 0676.46003)]. By definition it is the coarsest topology \(b_ X\) such that every closed ball of \(X\) is a closed set in \(b_ X\). \(X\) is said to have the \textit{ball-covering property} if the unit sphere of \(X\) can be covered by a countable collection of balls that do not contain the origin. It is shown that \(X\) has the ball-covering property if and only if every point of \(X\) is a \(G_\delta\) set in the ball topology. Some characterizations of smoothness and uniform smoothness of \(X\) in therms of ball-coverings of finite dimensional subspaces are given.
scientific article; zbMATH DE number 5862738
scientific article