A quest for convergence: exploring series in non-linear environments

Abstract

This note presents an extension of a result within the concept of \(\left[ \mathcal {S}\right] \)-lineability, originally due to Bernal-González, Conejero, Murillo-Arcila, and Seoane-Sepúlveda. Additionally, we provide a characterization of lineability in the context of complements of unions of closed subspaces in F-spaces in terms of \(\left[ \ell _{\infty }\right] \)-lineability. We also present a negative result in both normed spaces and p-Banach spaces. These findings contribute to the understanding of linearity within exotic settings in vector spaces.