Lineability and additivity in \(\mathbb R^{\mathbb R}\)
In recent years the question whether there exists an infinite dimensional vector space of functions \({\mathbb R}\to {\mathbb R}\) with given properties (e.g. nowhere differentiability or everywhere surjectivity) has been studied by various authors. A quantitative version of this question is expressed by the notion of \textit{lineability}: A familiy of functions \({\mathcal F}\subseteq {\mathbb R}^{\mathbb R}\) is called \(\kappa\)-lineable if it contains a vector space of dimension \(\kappa\) where \(\kappa\) is a cardinal number. In the present paper, the authors uncover a connection between lineability and another, seemingly unconnected notion, the \textit{additivity} of \({\mathcal F}\). The additivity of \({\mathcal F}\) is the cardinal number \({\mathcal A}({\mathcal F})\) satisfying \[ {\mathcal A}({\mathcal F}) = \min\left(\{\text{card}\,F:\;F\subseteq {\mathbb R}^{\mathbb R},\; \forall \varphi\in{\mathbb R}^{\mathbb R}:\;\varphi+F \not\subseteq {\mathcal F}\} \cup \{(2^c)^+\}\right) \] where \((2^c)^+\) is the successor cardinal of \(2^c\) with \(c\) as the cardinality of the continuum. In the first half of the paper, the authors prove that if \({\mathcal F}\) is such that \(\alpha {\mathcal F} \subseteq {\mathcal F}\) for all \(\alpha \in {\mathbb R}\) and if \(c\leq {\mathcal A}({\mathcal F}) \leq 2^c\), then \({\mathcal F}\) is \({\mathcal A}({\mathcal F})\)-lineable. This is the previously unknown connection between lineability and additivity. The authors also give several applications of this result (e.g., for everywhere surjective or for almost continuous functions). In the second half of the paper, the authors investigate a specific class \(J({\mathbb R})\) of ``pathological'' functions, the so-called \textit{Jones functions}. These are functions \(f:{\mathbb R}\to{\mathbb R}\) which have graph\(\,f \cap K \neq \emptyset\) for every closed set \(K\subseteq {\mathbb R}^2\) with uncountable projection on the \(x\) axis. The authors give a characterization of Jones functions, they investigate their relations to other, similar classes of functions, and they compute the additivity and prove the lineability of \(J({\mathbb R})\).
scientific article; zbMATH DE number 5731683
scientific article