Editing Norm (mathematics) (section)

===Absolute-value norm===
{{redirect|Absolute-value norm|the commutative algebra concept|Absolute value (algebra)}}

The [[absolute value]]
<math>|x|</math>
is a norm on the  vector space formed by the [[real number|real]] or [[complex number]]s. The complex numbers form a [[dimension (vector space)|one-dimensional vector space]] over themselves and a two-dimensional vector space over the reals; the absolute value is a norm for these two structures.

Any norm <math>p</math> on a one-dimensional vector space <math>X</math> is equivalent (up to scaling) to the absolute value norm, meaning that there is a norm-preserving [[isomorphism]] of vector spaces <math>f : \mathbb{F} \to X,</math> where <math>\mathbb{F}</math> is either <math>\R</math> or <math>\Complex,</math> and norm-preserving means that <math>|x| = p(f(x)).</math>
This isomorphism is given by sending <math>1 \isin \mathbb{F}</math> to a vector of norm <math>1,</math> which exists since such a vector is obtained by multiplying any non-zero vector by the inverse of its norm.