$G \cap P$ is compact and discrete, hence finite. Let $x$ be any point of $G$; it is equal to a linear combination $\sum_{i=1}^p t_i a_i$ of the $a_i$ with real coefficients. For each integer $m > 0$, consider the point

it belongs to $G$, and since $0 \leq mt_i - [mt_i] < 1$, it lies in $P$. Hence, first, $x = z_1 + \sum_{i=1}^p [t_i] a_i$, so that $G$ is generated by $G \cap P$; and secondly, since $G \cap P$ is finite, there exist two distinct integers $h, k$ such that $z_h = z_k$, so that $(h - k)t_i = [ht_i] - [kt_i]$ less ($1 \leq i \leq p$), and therefore the $t_i$ are rational numbers.

(*) We recall (Chapter IV, § 8, no. 2) that, for each real number $x, [x]$ is the integral part of $x$, i.e., the largest rational integer $\leq x$.

Proposition 1 shows that the condition is necessary. Also it is sufficient, for if it is satisfied we can write $t_i = m_i/d$, where $d$ and the $m_i$ are integers ($1 \leq i \leq p$); $b$ is therefore a linear combination, with integer coefficients, of the $p$ points $(1/d)a_i$; hence $G$ is a subgroup of the discrete group generated by these $p$ points, and so is itself discrete.

The result of Proposition 1 can be expressed as follows: if $q$ points $x_i$ ($1 \leq i \leq q$) of a discrete subgroup $G$ of $\mathbf{R}^n$ form a system which is dependent with respect to $\mathbf{R}$, then they form a system which is dependent with respect to $\mathbf{Q}$. It follows immediately that the rational rank of a discrete subgroup of $\mathbf{R}^n$ is equal to its rank.

The Corollary to Proposition 1, applied to the case where the $a_i$ are the $n$ vectors $e_i$ of the canonical basis of $\mathbf{R}^n$, gives us the following proposition:

Proposition 2 (Kronecker). Let $\theta_1, \theta_2, \ldots, \theta_n$ be $n$ real numbers. In order that, for each $\varepsilon > 0$, there exist an integer $q$ and $n$ integers

such that where the left-hand side of at least one of these inequalities does not vanish, it is necessary and sufficient that at least one of the $\theta_i$ be irrational.

By virtue of the properties of groups isomorphic to $\mathbf{Z}^p$ recalled earlier, it is enough to show that $G$ is a subgroup of a discrete group generated by a free system of $p$ points. Now since $G$ is of rank $p$ there exists a free system of $p$ points $a_i$ ($1 \leq i \leq p$) of $G$ such that every $x \in G$ is equal to a linear combination $\sum_{i=1}^p t_i a_i$ of the $a_i$ with real coefficients; since $G$ is discrete, Proposition 1 shows that the $t_i$ are rational. Furthermore, Proposition 1 shows that $G$ is generated by a finite number of points; since these points are linear combinations of the $a_i$ with rational coefficients, there exists an integer $d$ such that they are linear combinations with integer coefficients of the $p$ points $(1/d)a_i = a'_i$. It follows that $G$ is a subgroup of the group generated by the $a'_i$.

Theorem 1 can also be proved without recourse to the theory of invariant factors (cf. Exercise 1).

Discrete subgroups of $\mathbf{R}^n$ which are of rank $n$ are also called lattices in $\mathbf{R}^n$.

Let $(x_p)_{p \in \mathbf{N}}$ be an infinite sequence of points of $G$ such that $x_p \neq o$ and $\lim_{p \to \infty} x_p = o$; by hypothesis such a sequence exists. Let $P$ be an open cube with centre $o$, containing the $x_p$. Let $k_p$ denote the largest integer $h > 0$ such that $hx_p \in P$ (since $P$ is a bounded box and $x_p \neq o$, the existence of $k_p$ follows from Archimedes' axiom). The points $k_p x_p$ lie in a compact set $\overline{P}$, hence the sequence $(k_p x_p)_{p \in \mathbf{N}}$ has a cluster point $a \in \overline{P}$. If $\| (k_p + 1)x_p - a \| \leq \varepsilon$, we have

and since $\lim_{p \to \infty} x_p = o$, it follows that $a$ is also a cluster point of the sequence $((k_p + 1)x_p)$, whose points belong to the closed set $\mathbf{C}P$, by definition of $k_p$; hence $a \in \overline{P} \cap \mathbf{C}P$ (the frontier of $P$, Fig. 5), which implies $a \neq o$. Moreover, since $G$ is closed, we have $a \in G$. Let $t$ be any real number; since $|tk_p - [tk_p]| < 1$, the relation $\| k_p x_p - a \| \leq \varepsilon$ implies that $\| [tk_p]x_p - ta \| \leq |t| \varepsilon + \| x_p \|$; since $\lim_{p \to \infty} x_p = o$, $ta$ is a

!Figure 5 Figure 5.

!Figure 6 Figure 6.

cluster point of the sequence ([tk_p]x_p); but the points of this sequence belong to G, and consequently ta ∈ G, since G is closed. This completes the proof.

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 $r - p$.

This follows from Theorem 2, and Theorem 1 of no. 1 applied to the discrete group $W \cap G$.

This is an immediate consequence of Corollary 1. Corollary 2 of Theorem 2 shows that a closed subgroup G of $\mathbf{R}^n$ is completely determined up to isomorphism by two integers $\geq 0$: its rank, which we denote by $r(G)$, and the dimension of the largest vector subspace contained in G, which we call the dimension of G and denote by $d(G)$.

The only conditions which these integers have to satisfy are the inequalities $0 \leq d(G) \leq r(G) \leq n$.

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:

Let us apply this characterization of points lying in the closure of a subgroup $G$ to the case of the subgroup $G$ generated by the $n$ vectors $e_j$ of the canonical basis $(1 \leq j \leq n)$ and by an arbitrary number $m$ of points $a_i$ $(1 \leq i \leq m)$ of $\mathbf{R}^n$. To say that $(u|e_j)$ is an integer for $1 \leq j \leq n$ means that the $n$ coordinates of $u$ are integers; therefore:

it is necessary and sufficient that there exist no finite sequence $(r_j)$ of $n$ integers, not all zero, such that each of the $m$ numbers $\sum_{j=1}^n a_{ij} r_j$ is an integer.

For if $G$ is dense in $\mathbf{R}^n$, that is if $\overline{G} = \mathbf{R}^n$, then $G^* = \{0\}$, and conversely.

In particular $(m = 1)$:

it is necessary and sufficient that there exist no relation of the form $\sum_{j=1}^n r_j \theta_j = h$, where the $r_j$ are $n$ integers not all zero, and $h$ is an integer [which implies, in particular, that the $\theta_j$ and the ratios $\theta_j / \theta_k$ ($j \neq k$) must be irrational].

We may interpret this result as follows: for each integer $q \in \mathbf{Z}$, let $x_q$ denote the point with coordinates $q \theta_j - [q \theta_j]$ ($1 \leq j \leq n$); then Corollary 2 gives a necessary and sufficient condition for the set of points $x_q$ to be dense in the cube which is the product of $n$ intervals $[0, 1]$ in the factors of $\mathbf{R}^n$.

The real number $(u|x + y)$ is an integer for all $x \in G_1$ and all $y \in G_2$ if and only if $(u|x)$ is an integer for all $x \in G_1$ and $(u|y)$ is an integer for all $y \in G_2$, because $(u|x + y) = (u|x) + (u|y)$; hence

for any two subgroups $G_1, G_2$ of $\mathbf{R}^n$. If now we suppose that $G_1$ and $G_2$ are closed, we have $(G_1^* + G_2^*)^* = G_1 \cap G_2$ by Proposition 6; hence, taking the associated subgroups and applying Proposition 6 again,

The product space $T^n$ (and, by abuse of language, the topological group $T^n$) is called the n-dimensional torus; by Proposition 4 of Chapter V, § 1, no. 2, $T^n$ is compact, connected and locally connected.

Furthermore, if $C$ denotes a closed cube of side 1 in $\mathbf{R}^n$, $T^n$ is homeomorphic to the quotient space of $C$ by the equivalence relation " $x_i \equiv y_i \pmod{1}$ ($1 \leq i \leq n$)" between the points $x = (x_i)$ and $y = (y_i)$ of $C$. Thus $T^n$ is formed from the cube $C$ by "identification of opposite faces".

For $T^n = (R/Z)^n$ is isomorphic to $R^n/Z^n$ (Chapter III, § 2, no. 9, Corollary to Proposition 26) and $Z^n$ is a discrete subgroup of $R^n$ (Chapter III, § 2, no. 6, Proposition 19).

It follows that the groups $R^p \times T^{n-p}$ are all locally isomorphic to $R^n$ ($0 \geq p \geq n$); in § 2, no. 2, we shall see that they are the only connected groups which have this property.