Exercise 8, § 1

Let E be a topological vector space over a non-discrete topological division ring K. In order that there should exist a neighbourhood of the point (0, 0) in $K \times E$ such that the mapping $(\lambda, x) \mapsto \lambda x$ should be uniformly continuous in this neighbourhood, it is necessary and sufficient that there exist a neighbourhood $V_0$ of 0 in E such that the sets $\lambda V_0$ form a fundamental system of neighbourhoods of 0 in E, where $\lambda$ varies in the set of elements $\neq 0$ of K. When K is a division ring with a non-discrete valuation and E is Hausdorff, show that the uniform structure of E is then metrisable.