Let $G$ be a compact group with finite-dimensional, real representations $\phi$ and $\psi$ on $V$ and $W$ respectively. (e.g. $V = \mathbb{R}^m$, $W = \mathbb{R}^n$.) Is it true that, as is the case for finite groups, the dimension of the intertwiner space of the two representations is equal to the inner product of the characters? That is, do we still have

$$ \dim \text{Hom}_G(V, W) = \int_G \text{d}\lambda(g) \text{Tr}(\phi(g))\text{Tr}(\psi(g)) $$ where $\lambda$ is the Haar measure on $G$?

As an aside, do the proposed representations always exists? Do they exist if I insist they are unitary/orthogonal?