Let $G$ be a compact Lie group and $\Phi :G \rightarrow GL(V)$ be a finite dimensional representation of $G$; I want to show that there exists an hermitian product on $V$ such that $\Phi$ is unitary, i.e. $\Phi(g)$ is unitary $\forall g \in G$.
Let $( \ , \ )_V$ be a random hermitian product on $V$; now we define $$ (u,v)= \int_G (\Phi(x)u, \Phi(x)v)_V dx . $$ Here the integral is with respect to the (normalized) Haar measure on G. How can I show that this product does the job? Maybe I have to use that in a compact Lie group the Haar measure is both left-invariant and right-invariant? I think that it shouldn't be hard to prove but I'm still struggling; thanks in advance.