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Norm (abelian group)

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In mathematics, specifically abstract algebra, if is an (abelian) group with identity element then is said to be a norm on if:

  1. Positive definiteness: ,
  2. Subadditivity: ,
  3. Inversion (Symmetry): .[1]

An alternative, stronger definition of a norm on requires

  1. ,
  2. ,
  3. .[2]

The norm is discrete if there is some real number such that whenever .

Free abelian groups

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An abelian group is a free abelian group if and only if it has a discrete norm.[2]

References

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  1. ^ Bingham, N.H.; Ostaszewski, A.J. (2010). "Normed versus topological groups: Dichotomy and duality". Dissertationes Mathematicae. 472: 4. doi:10.4064/dm472-0-1.
  2. ^ a b Steprāns, Juris (1985), "A characterization of free abelian groups", Proceedings of the American Mathematical Society, 93 (2): 347–349, doi:10.2307/2044776, JSTOR 2044776, MR 0770551