Let $\mathbb{H}$ be a Hilbert space and $\mathbb{M}$ be a closed subspace. Show that if $Q:\mathbb{H}\rightarrow{\mathbb{H}}/{\mathbb{M}}$ is the natural map ( That is $Q(x)=x+\mathbb{M}$ ), then $Q:\mathbb{M}^{\perp}\rightarrow \mathbb{H}/\mathbb{M}$ is an isometric isomorphism
(Functional analysis by John B Conway:Banach Spaces: Section 4)
So what we have to prove is that $\forall x\in \mathbb{M}^{\perp}, ||Q(x)||=||x||$
And it is always true that $||Q(x)||=||x+\mathbb{M}||\leq||x||$.
So I was looking for the other direction of the inequality. And I don't see how to involve the properties of $\mathbb{M}^{\perp}$.
Alternatively I considered the inner product $\langle\cdot,\cdot\rangle$ and tried to prove $||Q(x)||^2=\langle Q(x),Q(x)\rangle=\langle x,x\rangle=||x||^2$ but wasn't successful.
Any help would be appreciated.