Let $G$ be a compact Lie group, let $V$ be a finite-dimensional complex [vector space](/page/Vector%20Space), and let $\rho:G\to GL(V)$ be a continuous finite-dimensional complex representation. Then there exists a Hermitian [inner product](/page/Inner%20Product) $(\cdot,\cdot)_V$ on $V$ such that, for every $h\in G$ and all $v,w\in V$,

\begin{align*} (\rho(h)v,\rho(h)w)_V=(v,w)_V. \end{align*}

Equivalently, each operator $\rho(h)$ is unitary with respect to $(\cdot,\cdot)_V$.