Theorem 2

Every Hausdorff topological vector space E, of finite dimension n, over a complete non-discrete valued division ring K, is isomorphic to $K_s^n$; in fact, for every basis $(e_i)_{1 \leq i \leq n}$ of E on K, the linear mapping $(\xi_i) \mapsto \sum_{i=1}^n \xi_i e_i$ is an isomorphism of $K_s^n$ on E.

Proposition 2 of I, p. 12, implies that th. 2 is true for $n = 1$; we argue by induction on $n$. Let $H$ be the vector subspace of $E$ generated by $e_1, e_2, ..., e_{n-1}$; the induction hypothesis is that the mapping $(\xi_i)_{1 \leq i \leq n-1} \mapsto \sum_{i=1}^{n-1} \xi_i e_i$ is an isomorphism of $K_s^{n-1}$ on $H$. The subspace $H$, being isomorphic to a product of complete spaces, is complete (GT, II, § 3.5, prop. 10); hence it is closed in $E$ (GT, II, § 3.4, prop. 8). Let $D$ be the subspace $Ke_n$ complementary to $H$ in $E$; $E$ is the topological direct sum of $H$ and $D$ (I, p. 13, cor. 2), therefore the mapping

of $K_s^{n-1} \times K_s$ on $E$ is an isomorphism.

When $n > 1$ the hypothesis that $K$ is complete is essential for the validity of theorem 2. In fact, let $K$ be a non-complete valued division ring, and let $\hat{K}$ be its completion : for each $a \neq 0$ of $\hat{K}$ the set $K.a$ is everywhere dense in $\hat{K}$, since $x \mapsto xa$ is a homeomorphism of $\hat{K}$ on itself. If $a \notin K$, the subspace $K + Ka$ of the topological vector space $\hat{K}$ on $K$ is of dimension 2 on $K$, but it is not isomorphic to $K_s^2$ since every subspace of dimension 1 in $K + Ka$ is dense in $K + Ka$.