Let K be a field complete for a proper valuation (absolute value) v. It is classic that a finite-dimensional K-vector space E admits a unique Hausdorff topology making it a topological K-vector space, and that that topology is the “cartesian product topology” in the sense that for any basis c1 …, cn of E, is a topological isomorphism from Kn to E [1, Chap. I, § 2, no. 3; 2, Chap. VI, § 5, no. 2]. It follows readily that any multilinear mapping from Em to a Hausdorff topological K-vector space is continuous. In particular, any multiplication on E making it a K-algebra is continuous in both variables. If for some such multiplication E is a field extension of K, then by valuation theory the unique Hausdorff topology of E is given by a valuation (absolute value) extending v.