A note on free vector balleans

Abstract

A vector balleans is a vector space over $\mathbb{R}$ endowed with a coarse structure in such a way that the vector operations are coarse mappings. We prove that, for every ballean $(X, \mathcal{E})$, there exists the unique free vector ballean $\mathbb{V}(X, \mathcal{E})$ and describe the coarse structure of $\mathbb{V}(X, \mathcal{E})$. It is shown that normality of $\mathbb{V}(X, \mathcal{E})$ is equivalent to metrizability of $(X, \mathcal{E})$.