If $V$ is irreducible then dim $Hom(V, W)^G$ is the multiplicity of $V$ in $W$? From Fulton and Harris, p. 16,

If $V$ is irreducible then dim $Hom(V, W)^G$ is the multiplicity of $V$ in $W$ by Schur's lemma.

The above is for a finite group $G$.
I tried to prove this but can't quite finish it.
For a map $\varphi \in Hom(V, W)^G$, we know that $Ker(V)$ is trivial, because V is irreducible.
Therefore from Schur's lemma $\varphi(V)$  is an isomorphism onto its image and $\varphi = \lambda I$.
So, we have now that each $\varphi$ maps a 1-dimensional irreducible rep, $V$ onto itself in $W$, and the dimension of $Hom(V, W)^G$ tells us how many different one dimensional subspaces of $W$ it maps into, or the multiplicity of $V$ in $W$.
Is my argument basically correct? Also, does this say anything about the dimension of W? I think not, since W may not just be a direct sum of representations of $V$.
 A: Using the hints that runway44@'s suggested, I think the argument is roughly like so.
From a corollary of Schur's lemma, for finite groups a representation $V$ is decomposable into a direct sum of irreducible representations.
So we can take, $$W = V^1 \oplus ... V^i \oplus U^1\oplus ...U^j$$
where the $V^i$ are isomorphic representations of $V$ and the $U^j$ are any other representations.
Now, as mentioned in the comments above,
$$Hom(V,V^1 \oplus ... V^i \oplus U^1\oplus ...U^j) \cong Hom(V,V^1) \oplus 
...Hom(V,V^i) \oplus Hom(V,U^1) \oplus...\oplus Hom(V,U^j)$$
We get this via projection maps $\pi_V^i$ and $\pi_U^j$. This follows from the fact that  for $w \in W$, $w = v^1+...v^i+u^1+...u^j$, uniquely.
So, $\varphi_{V^i} = \pi_{V^i} \circ \varphi$.
Now if we restrict to $Hom(V,W)^G$, the subspace of $G$-linear maps,
$V$ is irreducible, therefore, $\varphi_{V^i}$ is an isomorphism onto its image or the trivial map.
And if it is an isomorphism, $\varphi_{V^i} = \lambda I$ from Schur's lemma.
And $\varphi_{U^i}$ are all the trivial rep.
Therefore, each map in $Hom(V,W)^G$, consists of choosing a $\lambda^i$ for each $V^i$ in the decomposition of $W$. Hence, the dimension of $Hom(V,W)^G$, is just the multiplicity of the $V^i$.
