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arXiv · 1902.10857

A note on asymptotically monotone basic sequences and well-separated sets

Cleon S. Barroso
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Abstract

We remark that if $X$ is an infinite dimensional Banach space then every seminormalized weakly null sequence in $X$ has an asymptotic monotone basic subsequence. We also observe that if $X$ contains an isomorphic copy of $\ell_1$, then for every $\varepsilon>0$ there exist a $(1 +\varepsilon)$-equivalent norm $\vertiii{\cdot}$ on $X$ such that the unit sphere $(S_{(X, \vertiii{\cdot})})$ contains a normalized bimonotone basic sequences which is symmetrically $2$-separated.

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Cleon S. Barroso. 2019-02-28. A note on asymptotically monotone basic sequences and well-separated sets. https://arxiv.org/abs/1902.10857

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