Additivity and lineability in vector spaces

Abstract

Gámez-Merino, Munoz-Fernández and Seoane-Sepúlveda proved that if additivity $\mathcal{A}(\mathcal{F})>\mathfrak{c}$, then $\mathcal{F}$ is $\mathcal{A}(\mathcal{F})$-lineable where $\mathcal{F}\subseteq {\mathbb{R}}^{\mathbb{R}}$. They asked if $\mathcal{A}(\mathcal{F})>\mathfrak{c}$ can be weakened. We answer this question in negative. Moreover, we introduce and study the notions of homogeneous lineability number and lineability number of subsets of linear spaces.

9 pages

References in corpus (2)

• Large free linear algebras of real and complex functions
• An undecidable case of lineability in R^R

Cited by in corpus (1)

• Lineability, spaceability, and additivity cardinals for Darboux-like functions