Let $V$ be a vector space and $W$ be the dual space of $V$. For a subspace $U$ of $V$, we define $$ U^0 = \{f\in W \mid f(u)=0 \text{ for all } u\in U \}. $$
We are looking for an example in which $$ U_1^0 + U_2^0 \subset (U_1 \cap U_2)^0 $$ but not equal. We could prove the equality when $\dim(V)$ is finite. Thus, we are looking for an example for infinite case.