Proposition 2

Every Hausdorff topological vector space E of dimension 1 over a non-discrete valued division ring K is isomorphic to $K_s$; in fact, for every $a \neq 0$ in E, the mapping $\xi \mapsto \xi a$ of $K_s$ on E is an isomorphism (in other words every linear mapping of* $K_s$ on E is an isomorphism).

As the mapping $\xi \mapsto \xi a$ of $K_s$ on E is bijective and continuous (I, p. 1, def. 1), it is sufficient to show that it is bicontinuous. Let $\alpha$ be a real number $> 0$, we show that there exists a neighbourhood V of 0 in E such that if $\xi a \in V$ then $|\xi| < \alpha$. As K is not discrete, there exists an element $\xi_0 \in K$ such that $0 < |\xi_0| < \alpha$; but, as E is Hausdorff, there is a neighbourhood V of 0 such that $\xi_0 a$ does not belong to V. We can suppose that V is balanced (I, p. 7, prop. 4). But then if $\xi a \in V$ and $|\xi| \geq |\xi_0|$ we have $|\xi_0 \xi^{-1}| \leq 1$, and $\xi_0 a = (\xi_0 \xi^{-1}) (\xi a) \in V$; since this last statement is false we see that $\xi a \in V$ implies $|\xi| < |\xi_0| < \alpha$. This completes the proof.