Corollary 4

Let $E$ be a topological vector space over a complete non-discrete valued division ring. If $M$ is a closed vector subspace of $E$ and $F$ is a vector subspace of $E$ of finite dimension, then the subspace $M + F$ is closed in $E$. Write $\phi$ for the canonical homomorphism of $E$ on the quotient space $E/M$ (necessarily Hausdorff). Then the subspace $M + F$ is identical with $\overline{\phi}^{-1}(\phi(F))$. Now $\phi(F)$ is of finite dimension in $E/M$, therefore (cor. 1) $\phi(F)$ is closed in $E/M$, and, in consequence, $\overline{\phi}^{-1}(\phi(F))$ is closed in $E$.

We note that, if $M$ and $N$ are any two closed vector subspaces in a Hausdorff topological vector space $E$, then $M + N$ is not necessarily closed in $E$, \* even if $E$ is a Hilbert space \* (cf. IV, p. 64, exerc. 13, d)).