Let $G$ be a compact Lie group, let $\mathbb K\in\{\mathbb R,\mathbb C\}$, and let $V$ be a finite-dimensional [vector space](/page/Vector%20Space) over $\mathbb K$. If
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\begin{align*}
\rho:G\to GL(V)
\end{align*}
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is a smooth finite-dimensional representation of $G$, then there exist irreducible $\rho$-invariant subspaces $V_1,\dots,V_m\subset V$ such that