Exercise 7, § 1

Let $G$ be a closed subgroup of $\mathbf{R}^n$ and let $H$ be a closed subgroup of $G$. Then $G$ is the direct sum of $H$ and another closed subgroup $K$ if and only if $H$ is the intersection of $G$ and a vector subspace. (Use Exercise 5 for the necessity; to show sufficiency, apply Theorem 2 of no. 2 appropriately to $G$ and $H$, and use Exercise 3).