Isometry/Adjoint endomorphism/Example

For an isometry

${\displaystyle \varphi \colon V\longrightarrow V}$

on a Euclidean vector space, the inverse mapping ${\displaystyle {}\varphi ^{-1}}$ is the adjoint endomorphism. In this case, we have indeed

${\displaystyle {}\left\langle \varphi (v),w\right\rangle =\left\langle \varphi ^{-1}(\varphi (v)),\varphi ^{-1}(w)\right\rangle =\left\langle v,\varphi ^{-1}(w)\right\rangle \,.}$