On 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:\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, $c_0$).

Cited by in corpus (1)

• A problem on spreading models