Isometric mapping

A mapping from a metric space into a metric space preserving distances between points: If and , then

An isometric mapping is an injective mapping of a special type, indeed it is an immersion. If , that is, if is a bijection, then is said to be an isometry from onto , and and are said to be in isometric correspondence, or to be isometric to each other. Isometric spaces are homeomorphic. If in addition is the same as , then the isometric mapping is said to be an isometric transformation, or a motion, of .

If the metric spaces and are subsets of some topological space and if there exists a deformation such that is an isometric mapping from onto for each , then is called an isometric deformation of into .

An isometry of real Banach spaces is an affine mapping. Such a linear isometry is realized by (and called) an isometric operator.

Comments

The fact that isometries of real Banach spaces are affine is due to S. Ulam and S. Mazur [a1].

References

[a1]	S. Mazur, S. Ulam, "Sur les transformations isométriques d'espaces vectoriels" C.R. Acad. Sci. Paris , 194 (1932) pp. 946–948