the set of points that are annhilated by a subset $N\subset X'$ Let $X$ be a normed vector space over $\mathbb k$ ($\mathbb k=\mathbb R$ or $\mathbb C$). Let's consider $X'$ the set of continuous linear functionals $f:X\to \mathbb k$ called the dual of $X$. We define for each $M\subset X$, the set $Ann(M)\subset X'$ as the set of functionals $f$ such that $f(M)=0$. And also let's define for each $N\subset X'$ the set $ann(N)\subset X$ as the set of points that vanish on each $f\in N$.
I proved that $Ann(M)=${$0$} if and only if $\bar{M}=X$
I want to know if there is a similar result for the set $ann(N)=${$0$}
I don't know if the result it's true.
And maybe in a general view. how can we say about two sets if their $ann$ are the same? I know that if their $ Ann$ are the same, then they have the same closure.

It is not true that
$$
ann(N)=\{0\}\implies cl(N)=X^*
$$
For counterexample take $X=\ell_1$ and $N=c_0\subset X^*=\ell_\infty$. This is easy to see that $ann(N)=\{0\}$, but $cl(N)=N=c_0\subsetneq \ell_\infty$
