Let $U \subset V$, $\pi: V \rightarrow V/U$ be the quotient map. Show that for any $\phi \in (V/U)'$, $\pi ' (\phi) \in U^0$.
I know that since $\pi$ is a quotient map, it means that elements of $\pi(v) = v + U$ for every $v \in V$, I know that $\pi': (V/U)' \rightarrow V'$. We also know that the range of $\pi '$ is the annihilator of $U$, so maybe that by itself is enough? The hint is to show that $\phi \pi (v) = \phi(0 + U) = 0$, but I'm not quite sure how to put all this together.
If $u\in U$, then $\pi'(\phi)(u)=\phi\bigl(\pi(u)\bigr)=\phi(0)=0$.