Exercise 8, § 1

a) Let $H, K$ be two closed subgroups of $\mathbf{R}^n$. Show that if $r(H) + r(K)$ is equal to $r(H \cap K) + r(H + K)$, then $H + K$ is closed in $\mathbf{R}^n$. [Using Exercise 7 and the hypothesis given, reduce to the case where $H, K$ and $H \cap K$ all generate the same vector subspace; if $V$ (resp. $W$) denotes the largest subspace contained in $H$ (resp. $K$), show with the help of Exercise 5 that $V$ is generated by $V \cap K$, and $W$ is generated by $W \cap H$; finally, split up $H \cap K$ into the direct sum of $H \cap K \cap (V + W)$ and a discrete group, by using Exercise 7.]

b) Deduce from a) and Exercise 6 that, if $H$ and $K$ are two closed subgroups of $\mathbf{R}^n$ such that $H + K$ is a closed subgroup, then $H^* + K^*$ is also a closed subgroup.

c) Deduce from b) that if $H$ and $K$ are two closed subgroups of $\mathbf{R}^n$ such that $d(H) + d(K) = d(H \cap K) + d(H + K)$, then the subgroup $H + K$ is closed.