Norm (abelian group)

In mathematics, specifically abstract algebra, if is an (abelian) group with identity element then is said to be a norm on if:

1. Positive definiteness: ,
2. Subadditivity: ,
3. Inversion (Symmetry): .

An alternative, stronger definition of a norm on requires

1. ,
2. ,
3. .

The norm is discrete if there is some real number such that whenever .

Free abelian groups

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

References