Norm (abelian group)

In mathematics, specifically abstract algebra, if (G, +) is an abelian group then $\scriptstyle \nu \colon G\to \mathbb {R}$ is said to be a norm on the abelian group (G, +) if:

1. $\nu (g)>0\,\,\mathrm {if} \,\,g\neq 0$,
2. $\nu (g+h)\leq \nu (g)+\nu (h)$,
3. $\nu (mg)=|m|\nu (g)\,\,\mathrm {if} \,\,m\in \mathbb {Z}$.

The norm ν is discrete if there is some real number ρ > 0 such that ν(g) > ρ whenever g ≠ 0.

Free abelian groups

An abelian group is a free abelian group if and only if it has a discrete norm.