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 ${\displaystyle g\neq 0}$.

Free abelian groups

An abelian group is a free abelian group if and only if it has a discrete norm.[1]

References

1. Steprāns, Juris (1985), "A characterization of free abelian groups", Proceedings of the American Mathematical Society, 93 (2): 347–349, doi:10.2307/2044776, MR 0770551