# A note on free vector balleans

## TL;DR

This paper introduces the concept of free vector balleans, establishing their unique existence for any given ballean and linking their normality to the metrizability of the underlying space.

## Contribution

It defines free vector balleans, proves their uniqueness, and characterizes their normality in terms of metrizability.

## Key findings

- Existence and uniqueness of free vector balleans for any ballean.
- Normality of free vector balleans is equivalent to metrizability.
- Described the coarse structure of 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 $(X, \mathcal{E})$.