Bounded sequences having an even number of accumulation points

TL;DR

This paper proves that there is no infinite dimensional vector space of bounded sequences with an even number of accumulation points, answering a posed question negatively and showing such sequences are not lineable.

Contribution

It demonstrates that the set of sequences with an even number of accumulation points is not lineable, resolving a previously open question.

Findings

The set is not lineable in the space of bounded sequences.

Sequences with an even number of accumulation points cannot form an infinite dimensional vector space.

The result answers a specific open question in sequence space theory.

Abstract

In their papers, Leonetti, Russo, Somaglia, Menet and Papathanasiou posed the question of whether there exists an infinite dimensional vector space of sequences in $\ell_\infty$ having (except for the zero sequence) an even amount of accumulation points. Here we answer this question in the negative, by showing that this previous set of sequences is not even $3$-lineable and, therefore, not lineable.