Norm (abelian group) explained

In mathematics, specifically abstract algebra, if

(G, + ) is an (abelian) group with identity element e then \nu\colon G \to R is said to be a norm on (G, + ) if:

1. Positive definiteness

\nu(g) > 0 forall g \ne e and \nu(e) = 0 ,

1. Subadditivity

\nu(g + h) \le \nu(g) + \nu(h) ,

1. Inversion (Symmetry):

\nu( - g) = \nu(g) forall g \in G .^[1]

An alternative, stronger definition of a norm on

(G, + ) requires\nu(g) > 0 forall g \ne e ,\nu(g + h) \le \nu(g) + \nu(h) ,\nu(mg) = |m| \nu(g) forall m \in 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.

Notes and References

1. Bingham . N.H. . Ostaszewski . A.J. . Normed versus topological groups: Dichotomy and duality . Dissertationes Mathematicae . 2010 . 472 . 4. 10.4064/dm472-0-1 . free .