Editing Isometry (section)

==Generalizations==
* Given a positive real number ε, an '''ε-isometry''' or '''almost isometry''' (also called a '''[[Felix Hausdorff|Hausdorff]] approximation''') is a map <math>f \colon X \to Y </math> between metric spaces such that
*# for <math>x, x' \in X</math> one has <math>|d_Y(f(x),f(x')) - d_X(x,x')| < \varepsilon,</math> and
*# for any point <math>y \in Y</math> there exists a point <math>x \in X</math> with <math>d_Y(y, f(x)) < \varepsilon </math>

:That is, an {{mvar|ε}}-isometry preserves distances to within {{mvar|ε}} and leaves no element of the codomain further than {{mvar|ε}} away from the image of an element of the domain. Note that {{mvar|ε}}-isometries are not assumed to be [[continuous function|continuous]].

* The '''[[restricted isometry property]]''' characterizes nearly isometric matrices for sparse vectors.
* '''[[Quasi-isometry]]''' is yet another useful generalization.
* One may also define an element in an abstract unital C*-algebra to be an isometry: 
*:<math>a \in \mathfrak{A}</math> is an isometry if and only if <math>a^* \cdot a = 1.</math>
:Note that as mentioned in the introduction this is not necessarily a unitary element because one does not in general have that left inverse is a right inverse.

* On a [[pseudo-Euclidean space]], the term ''isometry'' means a linear bijection preserving magnitude. See also [[Quadratic form#Quadratic spaces|Quadratic spaces]].