### abstract

- We characterize topological (and uniform) spaces whose free (locally convex) topological vector spaces have a local $\mathfrak G $-base. A topological space $ X $ has a local $\mathfrak G $-base if every point $ x $ of $ X $ has a neighborhood base $(U_\alpha) _ {\alpha\in\omega^\omega} $ such that $ U_\beta\subset U_\alpha $ for all $\alpha\le\beta $ in $\omega^\omega $. To construct $\mathfrak G $-bases in free topological vector spaces, we exploit a new description of the topology of a free topological vector space over a topological (or more generally, uniform) space.