If $(\psi,V)$ and $(\omega,W)$ are two characters of two representations of a group $G$, then$$\langle \psi, \omega\rangle = \dim \operatorname{Hom}_G(V,W)?$$Here $\langle\cdot,\cdot\rangle$ is the standard inner product of characters of representations and $\operatorname{Hom}_G(V,W)$ is the vector space of intertwining operators from $V$ to $W$.
I am guessing this is true from examples I have seen. I think that Schur's lemma in fact says that this is true when $V,W$ are irreducible. But is it true in general? If so, how might one go about proving it? (I am not asking for a complete proof, just the basic idea of it.)
I should maybe add that as a definition the character of a representation is the trace thing.
