Norm (abelian group)
Encyclopedia
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:
The norm ν is discrete if there is some ρ > 0 such that ν(g) > ρ whenever g ≠ 0.
if and only if
it is discretely normed.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, specifically abstract algebra
Abstract algebra
Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...
, if (G, •) is an abelian group
Abelian group
In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...
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 groupFree abelian group
In abstract algebra, a free abelian group is an abelian group that has a "basis" in the sense that every element of the group can be written in one and only one way as a finite linear combination of elements of the basis, with integer coefficients. Hence, free abelian groups over a basis B are...
if and only if
If and only if
In logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....
it is discretely normed.