Geometry of Numbers
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1 Minkowski
-
Proof. If it is freely generated by \(\RR \)-linearly independent things, it is clearly discrete. If it is discrete, we can assume that the \(\RR \)-span of the elements is everything. Take the subgroup generated by by some \(\RR \)-basis in the subgroup to get a torus. By discreteness, the image of the subgroup in the quotient is finite, so the entire group must also be freely generated by \(\RR \)-linearly independent elements. This also proves the second statement. □
A lattice is a discrete subgroup with full \(\RR \)-rank. The area of a lattice \(L\) is the area of \(\RR ^n/L\), which is a torus, so is finite. It will be denoted \(A(L)\), and can be computed as the absolute value of the determinant of a basis.