stylistic:tone
Isometry with prototypical examples

If E is a Euclidean space, its associated vector space E → {\displaystyle {\overrightarrow {E}}} can be considered as a Euclidean space. Every point O ∈ E defines an isometry of Euclidean spaces

P ↦ O P →, ({\displaystyle P\mapsto {\overrightarrow {OP}},{\vphantom {\frac {(}{}}}}

which maps O to the zero vector and has the identity as associated linear map. The inverse isometry is the map

v ↦ O + v. {\displaystyle v\mapsto O+v.}

A Euclidean frame (O, e 1, …, e n) {\displaystyle (O,e_{1},\dots,e_{n})} allows defining the map

E → R n P ↦ (e 1 ⋅ O P →, …, e n ⋅ O P →), ({\displaystyle {\begin{aligned}E&\to \mathbb {R} ^{n}\\P&\mapsto {\Bigl (}e_{1}\cdot {\overrightarrow {OP}},\dots,e_{n}\cdot {\overrightarrow {OP}}{\Bigr)},{\vphantom {\frac {(}{}}}\end{aligned}}}

which is an isometry of Euclidean spaces. The inverse isometry is

R n → E (x 1 …, x n) ↦ (O + x 1 e 1 + ⋯ + x n e n). {\displaystyle {\begin{aligned}\mathbb {R} ^{n}&\to E\\(x_{1}\dots,x_{n})&\mapsto \left(O+x_{1}e_{1}+\dots +x_{n}e_{n}\right).\end{aligned}}}

This means that, up to an isomorphism, there is exactly one Euclidean space of a given dimension.

This justifies that many authors talk of R n {\displaystyle \mathbb {R} ^{n}} as the Euclidean space of dimension n.