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 \((X, \mathcal{E})\).