Corollary 1

Tag 0BYH, permanent. Cite it as [Bourbaki, Tag 0BYH].

General Topology, chapter VII, § 1. Subgroups and quotient groups of $\mathbf{R}^n$

In order that, for each $x = (x_j)$ $(1 \leq j \leq n)$ and each $\varepsilon > 0$ , there exist $m$ integers $q_i$ $(1 \leq i \leq m)$ and $n$ integers $p_j$ ( $1 \leq j \leq n$ ) such that

|q_1 a_{1j} + q_2 a_{2j} + \cdots + q_m a_{mj} - p_j - x_j| \leq \varepsilon \quad (1 \leq j \leq n)

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 $(m = 1)$ :

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

Cited by

Exercise 2, § 1

Cites

Nothing.