The following definition comes from Linear algebra (Hoffman)
Theorem $18$. If $S$ is any subset of a finite-dimensional vector space $V$, then $(S^0)^0$ is the subspace spanned by $S$
Proof. Let $W$ be the subspace spanned by $S$. Clearly $W^0=S^0$.
Therefore, what we are to prove is that $W=W^{00}$. By theorem 16 $$dimW+dimW^0=dimV\\dimW^0+dimW^{00}=dimV^*$$ and since $dimV=dimV^*$ we have $$dimW=dimW^{00}$$ Since $W$ is a subspace of $W^{00}$, we see that $W=W^{00}$
So my question is, how to show $W$ is a subspace of $W^{00}$?
Recall that, for a subset $E$ of $V^*$ we have $E^0 = \{\alpha \in V : f(\alpha)=0 \textrm{ for all } f \in E\}$. Now, choose $\alpha \in W$ and $f \in W^0$. Then, since any functional in $W^{0}$ vanishes on $W$, we have $f(\alpha) = 0$. Thus, we showed that $f(\alpha)=0$ for all $f \in W^0$, and then $\alpha \in (W^0)^0 = W^{00}$. Hence, $W \subseteq W^{00}$.