On asymptotic properties of Banach spaces under renormings

Edward Odell; Thomas Schlumprecht

On asymptotic properties of Banach spaces under renormings

Functional Analysis 2016-09-07 v1

Authors: Edward Odell , Thomas Schlumprecht

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, $c_0$ ).

Keywords

separable banach space equivalent norm weak compactness reflexivity spreading model isomorph

Cite

@article{arxiv.math/9709217,
  title  = {On asymptotic properties of Banach spaces under renormings},
  author = {Edward Odell and Thomas Schlumprecht},
  journal= {arXiv preprint arXiv:math/9709217},
  year   = {2016}
}

On asymptotic properties of Banach spaces under renormings

Abstract

Keywords

Cite

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