On Sequences with at Most a Finite Number of Zero Coordinates

TL;DR

This paper investigates the algebraic and topological structures of sequences with finitely many zero entries, extending classical results to new spaces and employing advanced notions like lineability.

Contribution

It introduces new insights into the structure of such sequences beyond classical \\ell_p\\ spaces, including the case where p is between 0 and 1, using notions like S-lineability.

Findings

The set of sequences with finitely many zeros is (alpha, c)-spaceable if and only if alpha is finite.

Extended classical results to include the case where p is between 0 and 1.

Verified spaceability properties in specific subspaces of \\ell_p\\.

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.