Every quotient of a reflexive space is reflexive

Question

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.

Answer 1

$\mathcal X$ reflexive, so every functional on $\mathcal X^*$ is an evaluation map. Now consider a functional $g$ on $(\mathcal X/M)^*$. One identifies $(\mathcal X/M)^*$ with the functionals on $\mathcal X$ that vanish on $M$. Now, extend $g$ to all of $\mathcal X^*$ by declaring it $0$ outside of $(\mathcal X/M)^*$. Now we get $x\in\mathcal X$ such that $g$ is the map $f\mapsto f(x)$

Then, restrict $g$ to just the functionals that vanish on $M$.

This gives rise to the realisation of $g$ as an evaluation map.

Since $g$ was arbitrary. We're done $\square$