Basic sequences and spaceability in ℓpℓp spaces

Let X   be a sequence space and denote by Z(X)Z(X) the subset of X   formed by sequences having only a finite number of zero coordinates. We study algebraic properties of Z(X)Z(X) and show (among other results) that (for p∈[1,∞]p∈[1,∞]) Z(ℓp)Z(ℓp) 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∞)Z(ℓ∞). In addition to this, we also give a thorough analysis of the existing algebraic structures within the sets Z(X)Z(X) and X∖Z(X)X∖Z(X) and their algebraic genericities.