Editing Metric space (section)

=== Normed vector spaces ===
{{anchor|Norm induced metric|Relation of norms and metrics}}
{{Main|Normed vector space}}
A [[normed vector space]] is a vector space equipped with a ''[[norm (mathematics)|norm]]'', which is a function that measures the length of vectors.  The norm of a vector {{mvar|v}} is typically denoted by <math>\lVert v \rVert</math>.  Any normed vector space can be equipped with a metric in which the distance between two vectors {{mvar|x}} and {{mvar|y}} is given by
<math display="block">d(x,y):=\lVert x-y \rVert.</math>
The metric {{mvar|d}} is said to be ''induced'' by the norm <math>\lVert{\cdot}\rVert</math>.  Conversely,{{sfn|Narici|Beckenstein|2011|pp=47–66}} if a metric {{mvar|d}} on a [[vector space]] {{mvar|X}} is
* translation invariant: <math>d(x,y) = d(x+a,y+a)</math> for every {{mvar|x}}, {{mvar|y}}, and {{mvar|a}} in {{mvar|X}}; and
* [[Absolute homogeneity|{{visible anchor|absolutely homogeneous|Homogeneous metric}}]]: <math>d(\alpha x, \alpha y) = |\alpha| d(x,y)</math> for every {{mvar|x}} and {{mvar|y}} in {{mvar|X}} and real number {{math|α}};
then it is the metric induced by the norm
<math display="block">\lVert x \rVert := d(x,0).</math>
A similar relationship holds between [[seminorm]]s and [[pseudometric space|pseudometric]]s.

Among examples of metrics induced by a norm are the metrics {{math|''d''<sub>1</sub>}}, {{math|''d''<sub>2</sub>}}, and {{math|''d''<sub>∞</sub>}} on <math>\R^2</math>, which are induced by the [[Manhattan norm]], the [[Euclidean norm]], and the [[maximum norm]], respectively.  More generally, the [[Kuratowski embedding]] allows one to see any metric space as a subspace of a normed vector space.

Infinite-dimensional normed vector spaces, particularly spaces of functions, are studied in [[functional analysis]].  Completeness is particularly important in this context: a complete normed vector space is known as a [[Banach space]].  An unusual property of normed vector spaces is that [[linear transformation]]s between them are continuous if and only if they are Lipschitz.  Such transformations are known as [[bounded operator]]s.