A general lineability criterion for complements of vector spaces

Abstract

In 1931, Banach proved that, far from being exceptional objects, the Weierstrass functions form a residual set in the space $\mathcal{C}[0,1]$ of continuous functions. Later on, in 1966, V. I. Gurariy showed that, except for zero, there is an infinite-dimensional linear subspace of Weierstrass functions. This was the first example of \textit{lineability}. Over the last decade, this topic has attracted the continuous attention of the mathematical community, with a steady stream of papers being published, many of them in highly ranked mathematical journals. Several lineability criteria are known and applied to specific topological vector spaces. To paraphrase L. Bernal-González and M. O. Cabrera in [J. Funct. Anal. \textbf{266} (2014), 3997-4025], ``sometimes, such criteria furnish unified proofs of a number of scattered results in the related literature''. In this article, we provide a general lineability criterion in the context of complements of vector spaces.