Asymptotic properties of Banach spaces under renormings

Odell, E.; Schlumprecht, Th.

doi:10.1090/S0894-0347-98-00251-3

Asymptotic properties of Banach spaces under renormings
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by E. Odell and Th. Schlumprecht

J. Amer. Math. Soc. 11 (1998), 175-188

DOI: https://doi.org/10.1090/S0894-0347-98-00251-3

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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, $c_{0}$).

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