Editing Isometry (section)

== Introduction ==

Given a metric space (loosely, a set and a scheme for assigning distances between elements of the set), an isometry is a [[Transformation (geometry)|transformation]] which maps elements to the same or another metric space such that the distance between the image elements in the new metric space is equal to the distance between the elements in the original metric space. 
In a two-dimensional or three-dimensional [[Euclidean space]], two geometric figures are [[Congruence (geometry)|congruent]] if they are related by an isometry;{{efn|
<p>'''3.11''' ''Any two congruent triangles are related by a unique isometry.''— Coxeter (1969) p.&nbsp;39<ref>{{harvnb|Coxeter|1969|page=39}}</ref></p>
}}
the isometry that relates them is either a rigid motion (translation or rotation), or a [[Function composition|composition]] of a rigid motion and a [[Reflection (mathematics)|reflection]].<!-- commentary: I presume "they" here means the geometric figures. Still commenting out because it doesn't seem to help. --><!--They are equal, up to an action of a rigid motion, if related by a [[Euclidean group#Direct and indirect isometries|direct isometry]] (orientation preserving).-->

Isometries are often used in constructions where one space is [[Embedding|embedded]] in another space. For instance, the [[Complete space#Completion|completion]] of a metric space <math>M </math> involves an isometry from <math>M </math> into <math>M',</math> a [[quotient set]] of the space of [[Cauchy sequence]]s on <math>M.</math> 
The original space <math>M </math> is thus isometrically [[isomorphism|isomorphic]] to a subspace of a [[complete metric space]], and it is usually identified with this subspace. 
Other embedding constructions show that every metric space is isometrically isomorphic to a [[closed set|closed subset]] of some [[normed vector space]] and that every complete metric space is isometrically isomorphic to a closed subset of some [[Banach space]].

An isometric surjective linear operator on a [[Hilbert space]] is called a [[unitary operator]].