= 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 form
$$
\omega((u_1,u_2),(v_1,v_2))=u_1v_2-u_2v_1
$$
is $SU_2$-invariant and identifies $V_1$ with $V_1^*$. Taking symmetric powers gives $V_n\cong V_n^*$ for every $n$. Therefore, if
$$
V\cong\bigoplus_n m_nV_n,
$$
then
$$
V^*\cong\bigoplus_n m_nV_n^*\cong V.
$$
This proves <self-duality of finite-dimensional SU2 representations>.
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