On Functions Whose Graph is a Hamel Basis, II

Płotka, Krzysztof

Canadian mathematical bulletin, Tome 52 (2009) no. 2, pp. 295-302

Résumé

We say that a function $h\,:\,\mathbb{R}\,\to \,\mathbb{R}$ is a Hamel function $(h\,\in \,\text{HF)}$ if $h$ , considered as a subset of ${{\mathbb{R}}^{2}},$ is a Hamel basis for ${{\mathbb{R}}^{2}}.$ We show that $\text{A}\left( \text{HF} \right)\,\ge \,\omega$ , i.e., for every finite $F\,\subseteq \,{{\mathbb{R}}^{\mathbb{R}}}$ there exists $f\,\in \,{{\mathbb{R}}^{\mathbb{R}}}$ such that $f\,+\,F\,\subseteq \,\text{HF}$ . From the previous work of the author it then follows that $\text{A}\left( \text{HF} \right)\,=\,\omega$ .

Détail

DOI : 10.4153/CMB-2009-032-x

Mots-clés : 26A21, 54C40, 15A03, 54C30, Hamel basis, additive, and Hamel functions