Norm (abelian group)

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

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

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