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In [[mathematics]], specifically [[abstract algebra]], if (''G'', +) is an [[abelian group]] then <math>\scriptstyle \nu\colon G \to \mathbb{R}</math> is said to be a '''norm on the abelian group''' (''G'', +) if: | |||
<ol> | |||
<li><math>\scriptstyle \nu(g) > 0</math> if <math>\scriptstyle g\ne 0</math>,</li> | |||
<li><math>\scriptstyle \nu(g+h) \le \nu(g) + \nu(h)</math>,</li> | |||
<li><math>\scriptstyle \nu(mg) = |m| \nu(g)</math> if <math>\scriptstyle m \in \mathbb{Z}</math>.</li> | |||
</ol> | |||
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.<ref>{{citation | |||
| last = Steprāns | first = Juris | |||
| doi = 10.2307/2044776 | |||
| issue = 2 | |||
| journal = Proceedings of the American Mathematical Society | |||
| mr = 770551 | |||
| pages = 347–349 | |||
| title = A characterization of free abelian groups | |||
| volume = 93 | |||
| year = 1985}}</ref> | |||
==References== | |||
{{reflist}} | |||
[[Category:Abelian group theory]] | |||
{{Abstract-algebra-stub}} | |||
Revision as of 15:06, 10 September 2013
In mathematics, specifically abstract algebra, if (G, +) is an abelian group then is said to be a norm on the abelian group (G, +) if:
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
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Template:Abstract-algebra-stub
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