Remark

— The results of Nos 2, 3 are no longer valid when $K$ is discrete. For example, let $K_1$ be a non-discrete valued division ring and $K$ be the discrete division ring obtained by endowing $K_1$ with the improper absolute value on $K_1$. Then $K_1$ is a topological vector space of dimension 1 over $K$, but it is not isomorphic to $K_s$. However, we can show that the results of Nos 2, 3 are valid even when $K$ is discrete, provided that we impose on the topological vector spaces considered, the property of having a fundamental system of balanced neighbourhoods of 0 (i.e. neighbourhoods $V$ such that $K.V = V$) (I, p. 27, exerc. 14); this condition (which is always satisfied when $K$ is a non-discrete valued division ring cf. I, p. 7, prop. 4) is not valid for all topological vector spaces over $K$ as the preceding example shows.