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Norm (mathematics)

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In mathematics, a norm is a function on a vector space that generalizes to vector spaces the notion of the distance from a point of a Euclidean space to the origin.

Formal definition of norm

Let X be a vector space over some subfield F of the complex numbers. Then a norm on X is any function having the following four properties:

  1. for all (positivity)
  2. if and only if x=0
  3. for all (triangular inequality)
  4. for all

A norm on X also defines a metric on X as . Hence a normed space is also a metric space.