I am working through Axler's Linear Algebra Done Right, where he uses the notation $U^0$ for the annihilator subspace of the dual space $V'$ such that if $\varphi \in U^0$, then $U \subseteq \text{null}(\varphi)$. Note that it is true that if $V$ is finite-dimensional, then $U^0 = V'$ implies that $U = \{0\}$. However, I am curious to find an example when $V$ is not finite-dimensional where $U$ needs not be equal to $\{0\}$. Thank you in advance for your help!
PS - if there is a proof that $U = \{0\}$ if $U^0 = V'$ when $V$ is infinite dimensional, then I would be interested in that too!