Exercise 6, § 1

Let $H$ and $K$ be two closed subgroups of $\mathbf{R}^n$ such that $H + K$ is a closed subgroup.

a) Show that, if $V$ (resp. $W$) is the largest vector subspace contained in $H$ (resp. $K$), then $V + W$ is the largest vector subspace contained in $G$, and therefore $d(H) + d(K) = d(H \cap K) + d(H + K)$ [note that the rational rank of $G/(V + W)$ is finite and consequently that $G$ cannot contain a line which is not contained in $V + W$].

b) Let $U$ be a subspace complementary to $V + W$, such that $G$ is the direct sum of $V + W$ and $M = G \cap U$, and such that $H \cap K$ is the direct sum of $(V + W) \cap (H \cap K)$ and $L = H \cap K \cap U$ (Exercise 5). Let $H'$ (resp. $K'$) be the subgroup of $M$ consisting of the components of the points of $H$ (resp. $K$) in the decomposition of $G$ into the direct sum of $V + W$ and $M$. Show that $M = H' + K'$ and that $H' \cap K' = L$.

c) If we put $H'' = H \cap (V + W)$ and $K'' = K \cap (V + W)$, show that $r(H'') + r(K'') = r(H'' \cap K'') + r(H'' + K'')$ (reduce to the case where $V \cap W = \{ o \}$, and show that then $H'' \cap K''$ is the direct sum of $V \cap K''$ and $W \cap H''$.

d) Show that

[use c) and Exercise 4, noting that $r(H) = r(H') + r(H'')$ and $r(K) = r(K') + r(K'')$].