Norm (abelian group)

In mathematics, specifically abstract algebra, if (G, +) is an abelian group then ν : G → R is said to be a norm on the abelian group (G, +) if:

1. ν ( g ) > 0 i f g ≠ 0 ,
2. ν ( g + h ) ≤ ν ( g ) + ν ( h ) ,
3. ν ( m g ) = | m | ν ( g ) i f m ∈ Z .

The norm ν is discrete if there is some real number ρ > 0 such that ν(g) > ρ whenever g ≠ 0.