- Norm (abelian group)
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In mathematics, specifically abstract algebra, if (G, •) is an abelian group then ν : G → ℝ is said to be a norm on the abelian group (G, •) if:
- ν(g) > 0 if g ≠ 0,
- ν(g • h) ≤ ν(g) + ν(h),
- ν(mg) = |m|ν(g) if m ∈ ℤ.
The norm ν is discrete if there is some ρ > 0 such that ν(g) > ρ whenever g ≠ 0.
Free abelian groups
It turns out that an abelian group is a free abelian group if and only if it is discretely normed.
References
- Juris Steprāns, A Characterization of Free Abelian Groups, Proceedings of the American Mathematical Society, (1985)
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