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I can prove that $U^0+W^0 \subset (U \cap W)^0$, however the other side I find very difficult to prove.

Surely I could take some linear functional $\phi \in (U \cap W)^0$, where $\phi(v) = 0 \space\forall v \in U \cap W$. This does not necessarily mean that $\phi(u) = 0 \space \forall u \in U!$

Am I thinking right?

Note: $X^0$ is the annihilator of $X$

ProfOak
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Governor
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    If $\phi\in(U\cap V)^0$, then $\phi\in U^0$ is not true in general. However, This question does not ask you to prove $\phi\in U^0$ or $\phi\in V^0$, it says $\phi\in U^0+V^0$. This assertion means there are $\psi\in U^0$ and $\eta\in V^0$ such that $\phi=\psi+\eta$. – fantasie Nov 24 '21 at 13:26
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    Check this question: https://math.stackexchange.com/questions/347480/the-annihilator-of-an-intersection-is-the-sum-of-annihilators – Ygor Arthur Nov 24 '21 at 13:39
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    Related: https://math.stackexchange.com/questions/3827386/annihilator-in-infinite-dimensional-vector-space – Kevin Aquino Nov 24 '21 at 14:07

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