Let $U,V$ be two infinite dimensional spaces over field $k$, does $U^*\cong V^*$ imply $U\cong V$?
The other direction of the question is trivial and can be found here.
Since $U\hookrightarrow U^{**}$ and $U^{**}\cong V^{**}$, one can treat both $U$ and $V$ as subspaces of $U^{**}$. I have no idea what to do next?