Definition:Isometry (Metric Spaces)

Definition

Let $M_1 = \left({A_1, d_1}\right)$ and $M_2 = \left({A_2, d_2}\right)$ be metric spaces.

Let $\phi: M_1 \to M_2$ be a bijection such that:

$\forall a, b \in M_1: d_1 \left({a, b}\right) = d_2 \left({\phi \left({a}\right), \phi \left({b}\right)}\right)$

Then $\phi$ is called an isometry.

Isometry Into

When $\phi: M_1 \to M_2$ is not actually a surjection, but satisfies the other conditions for being an isometry, then $\phi$ can be called an isometry into $M_2$.

Caution

Some sources do not insist that an isometry be surjective.

Make sure to know which prerequisites are used when quoting results about isometries.

Sources

• Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (1970)... (previous)... (next): $\text{I}: \ \S 5$: Complete Metric Spaces
• W.A. Sutherland: Introduction to Metric and Topological Spaces (1975): Definition $2.4.9$
• Thomas A. Whitelaw: An Introduction to Abstract Algebra (1978)... (previous)... (next): $\S 34 \ (5)$
• John B. Conway: A Course in Functional Analysis (1990) $\S I.5$