An isometry (also known as isometric embedding) is a map preserving the edistance between emetric spaces, or equivalently the distance between metric space.
On metric spaces, a map is an isometry if and only if it preserves distances.
An isometry preserves edistances.
An isometry preserves distances.
An isometry is injective
Any map on a subsingleton is an isometry
The identity is an isometry
The composition of isometries is an isometry
An isometry is an embedding
An isometry is continuous.
The inverse of an isometry is an isometry.
Isometries preserve the diameter
The injection from a subtype is an isometry
An isometry preserves the diameter in metric spaces
α and β are isometric if there is an isometric bijection between them.
An isometry induces an isometric isomorphism between the source space and the range of the isometry.
A metric space can be embedded in l^∞(ℝ) via the distances to points in a fixed countable set, if this set is dense. This map is given in the next definition, without density assumptions.
The embedding map is always a semi-contraction.
When the reference set is dense, the embedding map is an isometry on its image.
Every separable metric space embeds isometrically in ℓ_infty_ℝ.
The Kuratowski embedding is an isometric embedding of a separable metric space in ℓ^∞(ℝ)
The Kuratowski embedding is an isometry
Version of the Kuratowski embedding for nonempty compacts