I am working on a question where I have to show that for a Hilbert space $\mathscr{H}$, and closed linear subspace $Y \subset \mathscr{H}$, $\mathscr{H}/Y$ is a Hilbert space, isomorphic to $Y^{\bot}$.
I have already shown that $\mathscr{H}/Y$ is a Banach space, and I have defined a map $T: Y^{\bot} \to \mathscr{H}/Y : x \to [x]$. I have shown that $T$ is linear and bijective.
However, I have failed to come up with an inner product on $\mathscr{H}/Y$. Now I was wondering: can I define the inner product $([x],[y])$ on $\mathscr{H}/Y$ to be the same as the inner product between the originals $(x,y)$ in the space $Y^{\bot}$? I thought so, but I have no idea how to prove it... Any help would be much appreciated!
You need the fact that $$\mathscr{H} = Y \oplus Y^{\bot}$$ as vector spaces. So, if you want to define an inner product on $\mathscr{H}/Y \cong Y^{\bot}$ simply consider the inner product on $Y^{\bot}$. Precisely define $$([x],[y]) = (\pi(x), \pi(y))$$ where $\pi:\mathscr{H} = Y \oplus Y^{\bot} \longrightarrow Y^{\bot}$ is the projection on the second coordinate, and show that it is well defined.