Lemma 1

Let E and F be two topological vector spaces over a non-discrete valued division ring K, and let u be a continuous linear mapping of E in F such that the image of E is not meagre. Then, for every neighbourhood V of 0 in E, the set $\overline{u(V)}$ is a neighbourhood of 0 in F.

Let W be a balanced neighbourhood of 0 in E such that W + W ⊂ V (I, § 1.5, prop. 4). Let α be an element of K such that |α| > 1; then E is the union of the sets $\alpha^n W$ where n varies in $\mathbf{N}$; in fact, for all x ∈ E, there exists β ∈ K such that x ∈ βW (I, p. 7, prop. 4) and there exists an integer n ≥ 0 such that |β| < |α|^n, then x ∈ $\alpha^n W$ since W is balanced. Hence, u(E) is the union of the sequence of sets $u(\alpha^n W) = \alpha^n u(W)$, and as u(E) is not meagre in F, one at least of the sets $\alpha^n \overline{u(W)}$ possesses an interior point (GT, IX, § 5.3, def. 2) and therefore $\overline{u(W)}$ has an interior point.

Let $y_0$ be an interior point of $\overline{u(W)}$; since $-u(W) = u(W)$, and therefore $-\overline{u(W)} = \overline{u(W)}$ it follows that $0 = y_0 + (-y_0)$ is an interior point of $\overline{u(W)} + \overline{u(W)}$. As vector addition is a continuous mapping of $F \times F$ in $F$, the set $\overline{u(W)} + \overline{u(W)}$ is contained in the closure of the set

hence $\overline{u(V)}$ is a neighbourhood of 0 in $F$.

Before proving that (iv) implies (i) we prove the following lemma, where we make the convention that, in all metric spaces, $B_r(x)$ denotes the closed ball of centre $x$ and radius $r$.