Let $k$ be a field, let $V$ be a finite-dimensional $k$-vector space, and let $U \subset V$ be a linear subspace. Define the annihilator of $U$ in $V^*$ by
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\begin{align*}
U^0=\{\lambda \in V^* : \lambda(u)=0 \text{ for every } u \in U\}.
\end{align*}
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Define the annihilator of $U^0$ in $V^{**}$ by
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\begin{align*}
(U^0)^0=\{\Phi \in V^{**} : \Phi(\lambda)=0 \text{ for every } \lambda \in U^0\}.
\end{align*}
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Under the evaluation isomorphism
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\begin{align*}
\operatorname{ev}: V &\to V^{**}
\end{align*}
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given by $\operatorname{ev}(v)(\lambda)=\lambda(v)$ for every $v \in V$ and every $\lambda \in V^*$, one has