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:

$\nu (g)>0\,\,{\mathrm {if}}\,\,g\neq 0$ ,
$\nu (g+h)\leq \nu (g)+\nu (h)$ ,
$\nu (mg)=|m|\nu (g)\,\,{\mathrm {if}}\,\,m\in {\mathbb {Z}}$ .

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

↑ 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

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