Basic sequences and spaceability in l(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 is an element of [1, infinity]) Z(l(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(l(infinity)). In addition to this, we also give a thorough analysis of the existing algebraic structures within the sets Z(X) and X \ Z(X) and their algebraic genericities.