Isomorphism between a quotient space and a subspace

Question

When $L$ is an unbounded operator on a Hibert space $H$, and $K = Ker L$, $H/ K \simeq K^\perp$?

I'd appreciate it if you'd give me any help! Thank you.

**I've added the source below.

Answer 1

As already pointed $H/K$ may not be defined when $L$ is not bounded.

For any close subspace $K$ of $H$ we always have $H/K$ isometrically isomorphic to $K^{\perp}$. To get an isometrically isomorphism write any element $h$ uniqely as $h_1+h_2$ with $h_1 \in K$ and $h_2 \in K^{\perp}$. Verify that the map $h+K \to h_2$ is an isometric isomorphism.