General criteria for a strong notion of lineability

TL;DR

This paper introduces general criteria for the concept of -spaceability in vector spaces, extending existing results and exploring more nuanced notions of lineability and spaceability.

Contribution

It provides new general criteria for (,)-spaceability and applies these to extend recent findings in the field.

Findings

Established criteria for (,)-spaceability

Extended recent results on lineability and spaceability

Enhanced understanding of subtle distinctions in the concepts

Abstract

A subset $A$ of a vector space $X$ is called $\alpha$-lineable whenever $A$ contains, except for the null vector, a subspace of dimension $\alpha$. If $X$ has a topology, then $A$ is $\alpha$-spaceable if such subspace can be chosen to be closed. The vast existing literature on these topics has shown that positive results for lineability and spaceability are quite common. Recently, the stricter notions of $\left( \alpha,\beta\right) $% -lineability/spaceability were introduced as an attempt to shed light to more subtle issues. In this paper, among other results, we prove some general criteria for the notion of $\left( \alpha,\beta\right) $-spaceability and, as applications, we extend recent results of different authors.