In mathematics, specifically abstract algebra, if is an (abelian) group with identity element then is said to be a norm on if:
- Positive definiteness: ,
- Subadditivity: ,
- Inversion (Symmetry): .[1]
An alternative, stronger definition of a norm on requires
- ,
- ,
- .[2]
The norm is discrete if there is some real number such that whenever .
Free abelian groups
[edit]An abelian group is a free abelian group if and only if it has a discrete norm.[2]