I am trying to construct an isomorphic map between $H/ K$ and $K^\perp$.
Specifically, $L$ is an unbounded operator on a Hibert space $H$, with kernel $K$. How could I show $H/ K \simeq K^\perp$?
Does the quotient space $H/ K$ also have its inherent inner product? If so, does the isomorphic map preserve the inner products between $H/ K$ and $K^\perp$? ($K^\perp$ has its own inner product.)
I'd appreciate it if you'd give me any help! Thank you.