Proposition 3

Let $E$ be a topological vector space over a complete non-discrete valued division ring $K$. Let $M$ be a closed vector subspace of finite codimension $n$ in $E$. Then every subspace $N$ that is an algebraic complement of $M$ in $E$ is also a topological complement.

In $N$, the set $\{0\}$ is closed, since it is the intersection of $N$ and the set $M$ which is closed in $E$; thus $N$ is Hausdorff. As $E/M$ is also Hausdorff, the canonical mapping of $N$ on $E/M$, which is linear and bijective, is bicontinuous (I, p. 14, cor. 2), from which the proposition follows.