Duality $(M/N)^*\equiv N^\perp/M^\perp$ for closed subspaces $N\subset M$ of a Banach space

Question

Let $M$ be a closed subspace of a Banach space $X$. Then we can identify $(X/M)^*$ with $M^\perp$ and $M^*$ with $X^*/M^\perp$.

Indeed, if $Q^*:X\to X/M$ is the quotient map, then $Q^*:M^*\to X^*$ is a linear isometry with range $M^\perp$. Moreover, if $J:M\to X$ is the embedding, then $J^*:X^*\to M^*$ has kernel $M^\perp$ and induces a surjective linear isometry $\widehat{J^*}:X^*/M^\perp\to M^*$.

Let $M$ and $N$ be closed subspaces of a Banach space $X$ with $N\subset M$. Then we can identify $(M/N)^*$ and $N^\perp/M^\perp$, I think.

Which operator induces the identification a similar way as in the above cases $N=\{0\}$ and $M=X$?

Added June 17: Is there a reference for this identification?

cross-posted to m.se

Answer 1

As is so often the case, the isomorphism is the only reasonable map you can write down in the general case:

$N^\perp / M^\perp \to (M/N)^\ast, f+M^\perp \mapsto (m+N \mapsto f(m))$

All that's left to prove is that it actually works ;-)