Norm (abelian group)

In mathematics, specifically abstract algebra, if formula_1 is an abelian group then formula_2 is said to be a norm on formula_1 if:
The norm formula_7 is discrete if there is some real number formula_8 such that formula_9 whenever formula_10.
Free abelian groups.
An abelian group is a free abelian group if and only if it has a discrete norm.