Editing Normed vector space (section)

==Definition==
{{See also|Seminormed space}}

A '''normed vector space''' is a [[vector space]] equipped with a [[Norm (mathematics)|norm]]. A '''{{visible anchor|seminormed vector space}}''' is a vector space equipped with a [[seminorm]].

A useful [[Triangle inequality#Reverse triangle inequality|variation of the triangle inequality]] is
<math display=block>\|x-y\| \geq | \|x\| - \|y\| |</math> 
for any vectors <math>x</math> and <math>y.</math>

This also shows that a vector norm is a ([[Uniform continuity|uniformly]]) [[continuous function]].

Property 3 depends on a choice of norm <math>|\alpha|</math> on the field of scalars. When the scalar field is <math>\R</math> (or more generally a subset of <math>\Complex</math>), this is usually taken to be the ordinary [[absolute value]], but other choices are possible. For example, for a vector space over <math>\Q</math> one could take <math>|\alpha|</math> to be the [[p-adic absolute value|<math>p</math>-adic absolute value]].