Basic sequences and spaceability in ${\ell }_p$ spaces

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.

arXiv comment17 pages