Proposition 4

The subgroup $G^*$ associated with a subgroup $G$ of $\mathbf{R}^n$ is closed, and we have $(\overline{G})^* = G^*$.

For each $x \in G$, let $f_x(u)$ denote $(u|x)$; $f_x$ is a linear form, therefore continuous. Since $G^*$ is the intersection of the sets $f_x^{-1}(z)$ as $x$ runs through $G$, and since each of these sets is closed, it follows that $G^*$ is closed. On the other hand, if $u \in G^*$, we have $(u|x) \in \mathbf{Z}$ for all $x \in G$, and therefore, as $\mathbf{Z}$ is closed in $\mathbf{R}$, $(u|y) \in \mathbf{Z}$ for all $y \in \overline{G}$; thus $u \in (G)^*$. But we have $(\overline{G})^* \subset G^*$ (since $G \subset \overline{G}$), so that $(\overline{G})^* = G^*$.

Consider the structure of $G^*$ when $G$ is closed. By Corollary 1 to Theorem 2 of no. 2, there exists a base $(a_i)_{1 \leq i \leq n}$ of $\mathbf{R}^n$ such that $G$ coincides with the set of points

where the $t_i$ take all real values and the $n_j$ take all integer values. Hence $(u|x)$ is an integer for all these points $x$ if and only if $(u|a_i) = 0$ for $1 \leq i \leq p$ and $(u|a_i)$ is an integer for $p+1 \leq i \leq p+q$. Let us denote by $(a'_i)_{1 \leq i \leq n}$ the basis of $\mathbf{R}^n$ such that $(a'_i|a_j) = 0$ for $i \neq j$ and

(*) This notion is a particular case, corresponding to the group $\mathbf{R}^n$, of a general notion in the theory of duality of locally compact abelian groups (see, e.g., A. Weil, "L'integration dans les groupes topologiques et ses applications", Act. Sci. et Ind. no. 869, Paris, Hermann, 1950, pp. 108-109). The reader will observe the close analogy which exists between the properties of associated subgroups in $\mathbf{R}^n$ and those of orthogonal vector subspaces of a vector space and its dual.

[the basis "dual" to $(\mathbf{a}_i)$]; if we put $u = \sum_{i=1}^n u_i a'_i$, it is clear that the points $u \in G^*$ are characterized by the following conditions: $u_i = 0$ for $1 \leq i \leq p$, and $u_i \in \mathbf{Z}$ for

hence $G^*$ is the direct sum of a vector subspace $W$ with a basis consisting of the $a'_i$ such that $p + q + 1 \leq i \leq n$, and a discrete subgroup generated by the $a'_i$ such that $p + 1 \leq i \leq p + q$. In other words: