Isometry

:For the mechanical engineering and architecture usage, see isometric projection.

Related Topics:
Mechanical engineering - Architecture - Isometric projection

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In mathematics, an isometry, isometric isomorphism or congruence mapping is a distance-preserving isomorphism between metric spaces. Geometric figures which can be related by an isometry are called congruent.

Related Topics:
Mathematics - Distance - Isomorphism - Metric spaces - Congruent

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Isometries are often used in constructions where one space is embedded in another space. For instance, the completion of a metric space M involves an isometry from M into M', a quotient set of the space of Cauchy sequences on M. The original space M is thus isometrically 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 subset of some normed vector space and that every complete metric space is isometrically isomorphic to a closed subset of some Banach space.

Related Topics:
Embedded - Completion - Quotient set - Cauchy sequence - Complete metric space - Closed subset - Normed vector space - Banach space

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