Exercise 12, § 1

Let $K$ be a nondiscrete locally compact field, $E$ a left topological vector space over $K$. a) If $E$ is 1-dimensional, then $E$ is isomorphic to $K_s$ (imitate the proof of Prop. 2 of TVS, I, §2, No. 2, replacing the absolute value by the function $\mathrm{mod}_K$). b) If $E$ is $n$-dimensional, then $E$ is isomorphic to $K_s^n$ (imitate the proof of Th. 2 of TVS, I, §2, No. 3). c) Assume that $E$ is locally compact. Let $F$ be a linear subspace of $E$ of finite dimension $n$. For every $a \in K$, let $\mathrm{mode}_E(a)$ be the modulus of $x \mapsto ax$ in $E$. Since $F$ is closed in $E$ by b), one can form $\mathrm{mod}_{E/F}(a)$. Show that $\mathrm{mod}_E(a) = \mathrm{mod}_K(a)^n \mathrm{mod}_{E/F}(a)$. Show that if $\mathrm{mod}_K(a) < 1$ and $F \neq E$, then $\mathrm{mod}_{E/F}(a^n) < 1$. (One has $a^n \to 0$ as $n \to +\infty$; from this, deduce that $\mathrm{mod}_{E/F}(a^n) \to 0$ by arguing as in Prop. 12 of No. 10.) Deduce from this that $n \leq \log \mathrm{mode}(1/a)/\log \mathrm{mod}_K(1/a)$, hence that $E$ is finite-dimensional. (We shall see in CA, VI, that the topology of a locally compact field can be defined by an absolute value. The final result of c) will then be a special case of Th. 3 of TVS, I, §2, No. 4.)