Norm (abelian group)

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

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

The norm $\nu$ is discrete if there is some real number $\rho >0$ such that $\nu (g)>\rho$ whenever $g\neq 0$.

Free abelian groups

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