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 $(X, \mathcal{ E})$.
Categories: math.GN