 Solution

The classification of finite-dimensional representations of SU2 says that every irreducible is one of the $V_n$, and compactness gives complete reducibility. The standard alternating formis $S{U}_2$-invariant and identifies $V_1$ with $V_1^{\ast }$. Taking symmetric powers gives $V_n\cong V_n^{\ast }$ for every $n$. Therefore, ifthenThis proves self-duality of finite-dimensional SU2 representations.