# Definition:Isometry (Metric Spaces)/Definition 2

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## Definition

Let $M_1 = \struct {A_1, d_1}$ and $M_2 = \struct {A_2, d_2}$ be metric spaces or pseudometric spaces.

- $M_1$ and $M_2$ are
**isometric**if and only if there exist inverse mappings $\phi: A_1 \to A_2$ and $\phi^{-1}: A_2 \to A_1$ such that:

- $\forall a, b \in A_1: \map {d_1} {a, b} = \map {d_2} {\map \phi a, \map \phi b}$

- and:
- $\forall u, v \in A_2: \map {d_2} {u, v} = \map {d_1} {\map {\phi^{-1} } u, \map {\phi^{-1} } v}$

## Also known as

An **isometry** is also known as a **metric equivalence**.

Two **isometric spaces** can also be referred to as **metrically equivalent**.

## Also see

- Results about
**isometries**can be found here.

## Sources

- 1975: Bert Mendelson:
*Introduction to Topology*(3rd ed.) ... (previous) ... (next): Chapter $2$: Metric Spaces: $\S 7$: Subspaces and Equivalence of Metric Spaces: Definition $7.3$ - 2013: Francis Clarke:
*Functional Analysis, Calculus of Variations and Optimal Control*... (previous) ... (next): $1.2$: Linear mappings

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- 1990: John B. Conway:
*A Course in Functional Analysis*$\S I.5$