Norm (abelian group)

In mathematics, specifically abstract algebra, if (G, +) is an abelian group then $\scriptstyle \nu\colon G \to \mathbb{R}$ is said to be a norm on the abelian group (G, +) if:

1. $\nu(g) > 0 \,\,\mathrm{if}\,\, g\ne 0$,
2. $\nu(g+h) \le \nu(g) + \nu(h)$,
3. $\nu(mg) = |m| \nu(g) \,\,\mathrm{if}\,\, m \in \mathbb{Z}$.

The norm ν is discrete if there is some real number ρ > 0 such that ν(g) > ρ whenever g ≠ 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 770551