Set of weights of dual representation $V^*$ in terms of weights of $V$

Question

Let $\frak{g}$ be a finite dimensional semisimple Lie algebra and let $V$ be a $\frak{g}$-module. I've been trying to show that the set of weights $\Psi(V^*)$ of the dual module coincides with $-\Psi(V)$, the negative weights of our initial module $V$. It's eassy to do it whenever $V$ is finite dimensional and it's also relatively eassy to do it more generally, whenever $V$ is in the BGG category $\mathcal{O}$, in fact I think you only need $V$ to be $\frak{h}$-semisimple, where $\frak{h}$ is a Cartan of $\frak{g}$. My question is whether this is true for an arbitrary infinite dimensional module that might or might not be $\frak{h}$-semisimple. Proofs or counterexamples are welcome.