Let $G$ be a locally compact group, $X_1,X_2$ unitary irreps of $G$. Assume that there is a $G$-module isomorphism $\phi:X_1\to X_2$ (not necessarily continuous). Does this imply that $X_1\cong X_2$ as unitary representations of $G$?
In the case where $X_1$ and $X_2$ are finite dimensional, this is true because all linear maps between finite dimensional Hilbert spaces are continuous. If $G$ is compact, irreps are finite dimensional by the Peter-Weyl theorem.