Basic sequences and spaceability in $\ell_p$ spaces

TL;DR

This paper investigates the algebraic structure of sequences with finitely many zero entries in $\, ext{ell}_p$ spaces, proving that for all p, the set of such sequences does not contain infinite dimensional closed subspaces, resolving an open problem.

Contribution

It proves that $Z(\, ext{ell}_p)$ contains no infinite dimensional closed subspaces for all p, solving an open question about the linear structure of these sets.

Findings

$Z(\, ext{ell}_p)$ has no infinite dimensional closed subspaces for all p

Resolved an open question by Aron and Gurariy from 2003

Analyzed algebraic structures within $X \,\setminus Z(X)$ and their genericity

Abstract

Let $X$ be a sequence space and denote by $Z(X)$ the subset of $X$ formed by sequences having only a finite number of zero coordinates. We study algebraic properties of $Z(X)$ and show (among other results) that (for $p \in [1,\infty]$) $Z(\ell_p)$ does not contain infinite dimensional closed subspaces. This solves an open question originally posed by R. M. Aron and V. I. Gurariy in 2003 on the linear structure of $Z(\ell_\infty)$. In addition to this, we also give a thorough analysis of the existing algebraic structures within the set $X \setminus Z(X)$ and its algebraic genericity.