Sobre Noções Restritivas de Lineabilidade

Abstract

In this work, we gather and contextualize recent results on algebraic structures in nonlinear subsets of topological vector spaces, with emphasis on the notions of lineability, spaceability, and their stronger versions, such as (α, β)-lineability, (α, β)-spaceability, and the associated pointwise notions. Initially, we discuss general criteria for the strong notions of lineability and spaceability, establishing abstract conditions that explain the systematic failure of positive results when more restrictive requirements on dimension and closedness are imposed. We then present results on the set of continuous, integrable, and unbounded functions on [0,∞), showing that this set admits large-dimensional linear structures, including pointwise spaceability results, while also highlighting relevant limitations regarding (ℵ0,c)-spaceability. As applications, classical results in function and sequence spaces are recovered and generalized, revealing the delicate interplay between algebraic and topological properties in these contexts.