Theorem 1

Let E be a topological vector space over a non-discrete valued division ring. Let H be a hyperplane in E defined by the equation $f(x) = \alpha$ where f is a linear form not identically zero. Then H is closed in E if and only if f is continuous.

The condition is evidently sufficient (GT, I, § 2.1, th. 1); we show that it is necessary. We can suppose that H is a closed homogeneous hyperplane with the equation $f(x) = 0$. The quotient space E/H is then a Hausdorff topological vector space of dimension 1 on K. We can write $f = g \circ \phi$, where $\phi$ is the canonical mapping of E on E/H and g is a linear mapping of E/H on $K_s$; from prop. 2, g is continuous, thus the same is true of f.