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): .
An alternative, stronger definition of a norm on requires
, , .
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.
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