Annihilator of Subspace of Banach Space is Weak-* Closed

Theorem
Let $X$ be a Banach space.

Let $M$ be a vector subspace of $X$.

Let $X^\ast$ be the normed dual space of $X$.

Let $w^\ast$ be the weak-$\ast$ topology on $X^\ast$.

Let $M^\bot$ be the annihilator of $M$.

Then $M^\bot$ is closed in $\struct {X^\ast, w^\ast}$.