Isometric mapping

A mapping $f$ from a metric space $A$ into a metric space $B$ preserving distances between points: If $x, y \in A$ and $f \left({x}\right), f \left({y}\right) \in B$, then

$$ \rho_A \! \left({x, y}\right) = \rho_B \! \left({f \left({x}\right), f \left({y}\right)}\right). $$

An isometric mapping is an injective mapping of a special type, indeed it is an immersion. If $f \left({A}\right) = B$, that is, if $f$ is a bijection, then $f$ is said to be an isometry from $A$ onto $B$, and $A$ and $B$ are said to be in isometric correspondence, or to be isometric to each other. Isometric spaces are homeomorphic. If in addition $B$ is the same as $A$, then the isometric mapping is said to be an isometric transformation, or a motion, of $A$.

If the metric spaces $A_0$ and $A_1$ are subsets of some topological space $B$ and if there exists a deformation $F_t : A \to B$ such that $F_t$ is an isometric mapping from $A$ onto $A_t$ for each $t$, then $\left\{{A_t}\right\}$ is called an isometric deformation of $A_0$ into $A_1$.

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