A Quest for Convergence: Exploring Series in Non-Linear Environments

TL;DR

This paper extends the concept of [S]-lineability in topological vector spaces, providing new characterizations and negative results in normed and p-Banach spaces, advancing understanding of linear structures in complex environments.

Contribution

It introduces new characterizations of lineability in F-spaces and complements existing results, including negative findings in normed and p-Banach spaces.

Findings

Extended [S]-lineability results in F-spaces.

Characterization of lineability in complements of unions of closed subspaces.

Negative results in normed and p-Banach spaces.

Abstract

This note presents an extension of a result within the concept of [S]-lineability, originally developed in 2019 by L. Bernal-Gonz\'alez, J.A. Conejero, M. Murillo-Arcila, and J.B. Seoane-Sep\'ulveda . Additionally, we provide a characterization in terms of lineability in the context of complements of unions of closed subspaces in F-spaces, and finally, we present a negative result in both normed spaces and p-Banach spaces. These findings contribute to the understanding of linearity in exotic settings in topological vector spaces.