This is an announcement for the paper "Basic sequences and spaceability in $\ell_p$ spaces" by Daniel Cariello and Juan B. Seoane-Sepulveda.
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.
Archive classification: math.FA
Remarks: 17 pages
Submitted from: jseoane@mat.ucm.es
The paper may be downloaded from the archive by web browser from URL
http://front.math.ucdavis.edu/1307.2508
or