Remark 2

When the division ring of scalars K is the field \mathbf{R} or the field \mathbf{C}, every filter base \mathfrak{B} on E which satisfies just the two axioms (EV_I) and (EV_III) is a fundamental system of neighbourhoods of 0 for a topology compatible with the vector space structure of E. In fact, we need only prove that, in these conditions, for every $\lambda \neq 0$ in K and every $V \in \mathfrak{B}$ there exists $W \in \mathfrak{B}$ such that $\lambda W \subset V$. Now from (EV_III) there exists $W_1 \in \mathfrak{B}$ with $2\,W_1 \subset V$, and we deduce, inductively, that for every positive integer $n$, there exists $W_n \in \mathfrak{B}$ such that $2^n W_n \subset V$. As V is balanced, if we take $n$ so large that $2^n = |2^n| > |\lambda|$, then $W = W_n$ satisfies the condition, as required.

This result does not hold for every non-discrete valued division ring K, for in such a division ring it is no longer necessarily true that $|m\varepsilon| = m$ for every positive integer $m$ ($\varepsilon$ indicates the unit element of the division ring ; cf. I, p. 22, exerc. 1).