Unitary vector space/Isometry/Orthogonal complement/Fact/Proof

Proof

We have

${}U^{\perp }={\left\{v\in V\mid \left\langle v,u\right\rangle =0{\text{ for all }}u\in U\right\}}\,.$

For such a ${}v\in U^{\perp }$ and an arbitrary ${}u\in U$, we have

${}\left\langle \varphi (v),u\right\rangle =\left\langle \varphi ^{-1}(\varphi (v)),\varphi ^{-1}(u)\right\rangle =\left\langle v,u'\right\rangle =0\,,$

since ${}u'=\varphi ^{-1}(u)\in U$ due to the invariance of ${}U$. Therefore, ${}\varphi (v)\in U^{\perp }$.

To fact