Proposition 8

Let $E$ be a normed space over a non-discrete valued division ring, let $p$ be the norm on $E$, and let $\hat{E}$ be the additive topological group which is the completion of the additive group $E$. Then the function $(t, x) \to tx$ can be extended by continuity to $\hat{K} \times \hat{E}$ and defines on $\hat{E}$ a vector space structure over $\hat{K}$; the norm $p$ can be extended by continuity to a norm $\bar{p}$ on $\hat{E}$ which defines the topology of $\hat{E}$.

The extension of $tx$ by continuity is a particular case of the theorem of extension of a continuous bilinear mapping of a product of two abelian groups into a third (Chapter III, § 6, no. 5, Theorem 1); we have $1.x = x$ and $t(u x) = (t u)x$ for $t \in \hat{K}$, $u \in \hat{K}$ and $x \in \hat{E}$, by the principle of extension of identities; hence the external law $(t, x) \to t x$ indeed defines on $\hat{E}$ a structure of a vector space over $\hat{K}$. On the other hand, the invariant metric $\overline{d}(x, y) = \overline{p}(x - y)$ extends to an invariant metric $\overline{d}$ on $\hat{E}$ (§ 2, no. 1, Proposition 1) which defines the topology of $\hat{E}$; if we set $\overline{p}(x) = \overline{d}(0, x)$, then $\overline{p}$ is the extension of $p$ by continuity, and satisfies axioms (NO_I) and (NO_{II}); by virtue of the continuity of $t x$ on $\hat{K} \times \hat{E}$, $\overline{p}$ also satisfies (NO_{III}) (principle of extension of identities) and is therefore a norm on $\hat{E}$.

When we have to consider a definite normed space structure on a vector space $E$, we shall usually denote the norm of a vector $x$ by $||x||$, unless this notation is likely to lead to confusion.