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](https://math.stackexchange.com/questions/4173533/duality-m-n-equiv-n-perp-m-perp-for-closed-subspaces-n-subset-m-of-a-b)



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


