Theorem 3

Let $K$ be a complete non-discrete valued division ring. If $E$ is a Hausdorff topological vector space over $K$, which is such that some neighbourhood $V$ of 0 in $E$ is precompact, then $E$ is of finite dimension. If $E \neq \{0\}$, then both $K$ and $E$ are locally compact.

In proving the first assertion, we need consider only the case when $E$ is complete; for $E$ is an everywhere dense subspace of its completion $\hat{E}$, and the closure $\overline{V}$ of $V$ in $\hat{E}$ is compact and is a neighbourhood of 0 in $\hat{E}$ (GT, III, § 3.4, prop. 7).

We can suppose then that there is a compact neighbourhood $V$ of 0 in $E$. Let $\alpha \in K$ be such that $0 < |\alpha| < 1$; then there are finitely many points $a_i \in V$ such that

Let $M$ be the finite dimensional subspace of $E$ generated by the $a_i$; it is closed in $E$ (I, p. 14, cor. 1). In the Hausdorff topological vector space $E/M$ the canonical image of $V$ is a compact neighbourhood $W$ of 0, such that $W \subset \alpha W$; hence $\alpha^{-1} W \subset W$, and, by induction on $n$, $\alpha^{-n} W \subset W$ for every positive integer $n$. As $W$ is absorbent, we conclude that $W = E/M$; and thus $E/M$ is compact. To complete the proof of the first assertion in the theorem, it is sufficient, therefore, to establish the following lemma.