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

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 $\epsilon >0$ there exist a $(1+\epsilon )$-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.

9 pages, comments are more than welcome. Corrected version