%0 Journal Article
%T On asymptotic properties of Banach spaces under renormings
%A Edward Odell
%A Thomas Schlumprecht
%J Mathematics
%D 1997
%I arXiv
%X It is shown that a separable Banach space $X$ can be given an equivalent norm $|\!|\!|\cdot |\!|\!|$ with the following properties:\quad If $(x_n)\subseteq X$ is relatively weakly compact and $\lim_{m\to\infty} \lim_{n\to\infty}\break |\!|\!| x_m + x_n |\!|\!| = 2\lim_{m\to\infty} |\!|\!| x_m|\!|\!|$ then $(x_n)$ converges in norm. This yields a characterization of reflexivity once proposed by V.D.~Milman. In addition it is shown that some spreading model of a sequence in $(X, |\!|\!|\cdot |\!|\!|)$ is 1-equivalent to the unit vector basis of $\ell_1$ (respectively, $c_0$) implies that $X$ contains an isomorph of $\ell_1$ (respectively, $c_0$).
%U http://arxiv.org/abs/math/9709217v1