Consider a faithful representation of a compact Lie group $G$ on a (complex) vector space $V$. How can I construct a Hermitian inner product on $V$ which is invariant under action by $G$?
I.e. I want to construct a metric so that $G$ "acts like unitary matrices" on $V$. I think compactness of $G$ is somehow required.
Take any Hermitian inner product $(\cdot,\cdot)$ on $V$. Now, define a new inner product $\langle\cdot,\cdot\rangle$ on $V$ by$$\langle v,w\rangle=\int_G(gv,gw)\,\mathrm dg,$$where the integration is taken with respect to a Haar measure on $G$. The fact that $G$ is locally compact is sufficient to assure that such a measure exists and the compactness of $G$ assures that the map $g\mapsto(gv,gw)$ is continuous and bounded and therefore integrable. It is easy to check that $\langle\cdot,\cdot\rangle$ is a Hermitian inner product on $G$ invariant under the action of $G$.