Asymptotic properties of Banach spaces under renormings

Abstract:

It is shown that a separable Banach space $X$ can be given an equivalent norm $||| \cdot |||$ with the following properties: If $(x_{n})\subseteq X$ is relatively weakly compact and $\lim _{m\to \infty } \lim _{n\to \infty } ||| x_{m}+x_{n}||| = 2\lim _{m\to \infty } ||| x_{m}|||$, 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, $c_{0}$).