Recall that $S^0 = \{v \in V : \langle v, s \rangle = 0 \text{ for all } s \in S\}$ and $S^{00} = (S^0)^0$.
Let $s \in S$. We want to show that $s \in S^{00}$. Since $s \in S$, for any $v \in S^0$, we have $\langle v, s \rangle = 0$. This means that $s$ is
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