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

## TL;DR

This paper explores properties of basic sequences in Banach spaces, showing that weakly null sequences have asymptotic monotone subsequences and constructing well-separated sequences under certain conditions.

## Contribution

It demonstrates that every weakly null sequence in an infinite-dimensional Banach space has an asymptotic monotone basic subsequence and constructs well-separated sequences in spaces containing _1 with equivalent norms.

## Key findings

- Weakly null sequences have asymptotic monotone basic subsequences.
- Existence of well-separated basic sequences in spaces with _1.
- Construction of equivalent norms with specific geometric properties.

## 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.