Complements of unions: insights on spaceability and applications

Abstract

This paper presents two general criteria to determine spaceability results in the complements of unions of subspaces. The first criterion applies to countable unions of subspaces under specific conditions and is closely related to the results of Kitson and Timoney in [J. Math. Anal. Appl. \textbf{378} (2011), 680-686]. This criterion extends and recovers some classical results in this theory. The second criterion establishes sufficient conditions for the complement of a union of Lebesgue spaces to be $\left(\alpha,\beta\right)$-spaceable, or not, even when they are not locally convex. We use this result to characterize the measurable subsets having positive measure. Armed with these results, we have improved existing results in environments such as: Lebesgue measurable function sets, spaces of continuous functions, sequence spaces, nowhere H\"{o}lder function sets, Sobolev spaces, non-absolutely summing operator spaces, and even sets of functions of bounded variation.