Exercise 4, § 1

Let $E$ be a vector space over a division ring $K$ with a valuation; we suppose that $E$ carries a metrisable topology compatible with its additive group structure. Suppose further that this topology satisfies axioms (EVT'_I) and (EVT'_II); show that if one of the two metrisable groups $K, E$ is complete then the topology also satisfies (EVT'_III) and is, in consequence, compatible with the vector space structure of $E$ (cf. GT, IX, § 5, exerc. 23).