Norm (abelian group)

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


 * 1) Positive definiteness: $$ \nu(g) > 0 \text{ for all } g \ne e \text{ and } \nu(e) = 0$$,
 * 2) Subadditivity: $$ \nu(g+h) \le \nu(g) + \nu(h)$$,
 * 3) Inversion (Symmetry): $$ \nu(-g) = \nu(g) \text{ for all } g \in G$$.

An alternative, stronger definition of a norm on $$(G, +)$$ requires


 * 1) $$ \nu(g) > 0 \text{ for all } g \ne e$$,
 * 2) $$ \nu(g+h) \le \nu(g) + \nu(h)$$,
 * 3) $$ \nu(mg) = |m| \, \nu(g) \text{ for all } m \in \mathbb{Z}$$.

The norm $$\nu$$ is discrete if there is some real number $$\rho > 0$$ such that $$\nu(g) > \rho$$ whenever $$g \ne 0$$.

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