On $r$-reflexive Banach spaces

summary:A Banach space $X$ is called {\it $r$-reflexive\/} if for any cover $\Cal U$ of $X$ by weakly open sets there is a finite subfamily $\Cal V\subset\Cal U$ covering some ball of radius 1 centered at a point $x$ with $\|x\|\leq r$. We prove that an infinite-dimensional separable Banach space $X$ is $\infty$-reflexive ($r$-reflexive for some $r\in \Bbb N$) if and only if each $\varepsilon $-net for $X$ has an accumulation point (resp., contains a non-trivial convergent sequence) in the weak topology of $X$. We show that the quasireflexive James space $J$ is $r$-reflexive for no $r\in \Bbb N$. We do not know if each $\infty$-reflexive Banach space is reflexive, but we prove that each separable $\infty$-reflexive Banach space $X$ has Asplund dual. As a by-product of the proof we obtain a covering characterization of the Asplund property of Banach spaces