On Sequences with at Most a Finite Number of Zero Coordinates

Abstract: 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-Sep\'ulveda, our research reveals new insights and complements their notable results beyond the classical ( \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 ( \ell_p ) containing ( c_0 ), is (alpha, c)-spaceable if and only if alpha is finite.