On Sequences with at Most a Finite Number of Zero Coordinates

In this paper, we analyze the existence of algebraic and topological structures in the set of sequences that contain only a finite number of zero coordinates. Inspired by the work of Daniel Cariello and Juan B. Seoane-Sepúlveda, our research reveals new insights and complements their notable results beyond the classical ${\mathrm{\ell }}_p$ spaces for $p$ in the interval from 1 to infinity, including the intriguing case where $p$ is between 0 and 1. Our exploration employs notions such as S-lineability, pointwise lineability, and (alpha, beta)-spaceability. This investigation allowed us to verify, for instance, that the set $F\setminus Z(F)$, where $F$ is a closed subspace of ${\mathrm{\ell }}_p$ containing $c_0$, is (alpha, c)-spaceable if and only if alpha is finite.