Editing Absolute value (section)

===Vector spaces===
{{Main|Norm (mathematics)}}
Again the fundamental properties of the absolute value for real numbers can be used, with a slight modification, to generalise the notion to an arbitrary vector space.

A real-valued function on a [[vector space]]&nbsp;{{mvar|V}} over a field&nbsp;{{mvar|F}}, represented as {{math|{{norm}}}}, is called an '''absolute value''', but more usually a [[Norm (mathematics)|'''norm''']], if it satisfies the following axioms:

For all&nbsp;{{mvar|a}} in&nbsp;{{mvar|F}}, and {{math|'''v'''}}, {{math|'''u'''}} in&nbsp;{{mvar|V}},

:{| cellpadding=10
|-
|<math>\|\mathbf{v}\|  \ge 0 </math>
|Non-negativity
|-
|<math>\|\mathbf{v}\| = 0 \iff \mathbf{v} = 0</math>
|Positive-definiteness
|-
|<math>\|a \mathbf{v}\| = \left|a\right| \left\|\mathbf{v}\right\| </math>
|Absolute homogeneity or positive scalability
|-
|<math>\|\mathbf{v} + \mathbf{u}\| \le \|\mathbf{v}\| + \|\mathbf{u}\| </math>
|Subadditivity or the triangle inequality
|}

The norm of a vector is also called its ''length'' or ''magnitude''.

In the case of [[Euclidean space]] <math>\mathbb{R}^n</math>, the function defined by

:<math>\|(x_1, x_2, \dots , x_n) \| = \sqrt{\textstyle\sum_{i=1}^{n} x_i^2}</math>

is a norm called the Euclidean norm. When the real numbers <math>\mathbb{R}</math> are considered as the one-dimensional vector space <math>\mathbb{R}^1</math>, the absolute value is a [[Norm (mathematics)|norm]], and is the {{mvar|p}}-norm (see [[L^p space#Definition|L<sup>p</sup> space]]) for any&nbsp;{{mvar|p}}. In fact the absolute value is the "only" norm on <math>\mathbb{R}^1</math>, in the sense that, for every norm {{math|{{norm}}}} on <math>\mathbb{R}^1</math>, {{math|1={{norm|''x''}} = {{norm|1}} ⋅ {{abs|''x''}}}}.

The complex absolute value is a special case of the norm in an [[inner product space]], which is identical to the Euclidean norm when the complex plane is identified as the [[Euclidean plane]]&nbsp;<math>\mathbb{R}^2</math>.