Relevant definitions:
- Given a vector space $V$, $V'$ is its dual space.
- For a given subspace $U$, $U^{0} = \{\phi\in V' | \phi(u)=0 \forall u\in U\}$
The equality $(U\cap W)^{0} = U^{0}+W^{0}$ holds if $U,W$ are subspaces of a finite-dimensional $V$. Moreover, the inclusion $U^{0}+W^{0}\subseteq (U\cap W)^{0}$ holds regardless of the dimensionality of $V$.
Proof: Take $\phi_{1}\in U^{0}$ and $\phi_{2}\in W^{0}$. Hence, $\phi_{1}(u)=0$ for any $u\in U\cap W$ and $\phi_{2}(w)=0$ for any $w\in U\cap W$.
Is there a counterexample to the reverse inclusion $(U\cap W)^{0}\subseteq U^{0}+W^{0}$ where $V$ is infinite-dimensional? All proofs I've read of the finite-dimansional case leveraged some form of finite dimensionality instead of proving that there exist two functionals $\phi\in U^{0}$ and $\varphi\in W^{0}$ for every $f\in (U\cap W)^{0}$ s.t. $f=\phi + \varphi$, so I suspect there exists a counterexample.
