TITLE: Every quotient of a reflexive space is reflexive
QUESTION [5 upvotes]: How do you prove the following?

If $\mathcal{X}$ is reflexive and $M \leq \mathcal{X} \rightarrow \mathcal{X}/M$ is reflexive

There is no assumption that $\mathcal{X}$ is a Banach space.

REPLY [2 votes]: X is reflexive implies it is Banach and, as M is closed in X, so is X/M. Since duals are Banach always, $X^*$ and $(X/M)^*$ are Banach, which is a closed subspace of the dual of X. Now dual of X is reflexive and hence its closed subspaces are also reflexive. Infact, the quotient is reflexive.