Theorem 2

Let G be a closed subgroup of $\mathbf{R}^n$, of rank r ($0 \leq r \leq n$). Then there is a largest vector subspace V contained in G; for every vector subspace W complementary to V, $W \cap G$ is discrete and G is the direct sum of V and $W \cap G$.

Let us first establish the existence of V by showing that the union of all the lines through o which lie in G is a vector subspace: indeed, the vector subspace generated by the union of these lines is the same as the subgroup they generate. The group G is the direct sum of V and $W \cap G$; for if $x \in G$, we have $x = y + z$ where $y \in V$ and $z \in W$; since $V \subset G$, $z = x - y \in G$ and therefore $z \in W \cap G$. It remains to show that $W \cap G$ is discrete; this follows from Proposition 3, since $W \cap G$ is a closed subgroup which contains no lines, by reason of the definition of V.

If $G \neq V$, we may say that G is the union of a countable infinity of linear varieties, parallel to V and passing through the points of the discrete group $W \cap G$ (Fig. 6).

If $p$ is the dimension of the vector subspace V, we have $p \leq r$, and $W \cap G$ is a discrete group of rank $r - p$.