Euclidean vector space/Isometry with standard space/Fact/Proof

Proof

Let ${}u_{1},\ldots ,u_{n}$ be an orthonormal basis of ${}V$, and let

$\varphi \colon \mathbb {R} ^{n}\longrightarrow V$

be the linear mapping given by

${}\varphi (e_{i})=u_{i}\,.$

Because of fact  (3), this is an isometry.

To fact