Double annihilator of subspace of $X'$ is its weak*-closure

Question

Let $X$ be a Banach space with dual space $X'$. Let $N$ be a subspace of $X'$. Can anyone show me why the double annihilator of $N$ is its weak*-closure?

By double annihilator I mean:

annihilator $N^{\perp}=\{x\in X:\lambda(x)=0, \forall \lambda \in N\}$

double annihilator $(N^{\perp})^{\perp}=\{\lambda\in X':\lambda(x)=0, \forall x \in N^{\perp}\}$

Thank you in advance!

Answer 1

You can find a proof in this link under Proposition 8 (2).

https://math.la.asu.edu/~quigg/teach/courses/578/2008/notes/adjoints.pdf

Note that the Hahn-Banach theorem also works for locally convex spaces (here $X^*$ with the weak*-topology) and that a functional on $X^*$ is weak*-continuous if and only if it is an evaluation map $f \mapsto f(x)$ for some $x \in X$.