I am having trouble with the following exercise in character theory:
If $\chi, \psi, \zeta$ are irreducible characters of a finite group then $\langle \chi\psi, \zeta \rangle \leq \zeta(1)$.
I can show that $\langle \chi\psi, \zeta \rangle = \dim \textrm{Hom}_G(U, V\otimes_\mathbb{C} W)$ where V, W, U are the irreducible G-modules corresponding to $\chi, \psi, \zeta$ respectively, but it seems that I have merely translated the question into one about homomorphisms and tensor products. It doesn't help me much or at least I can't see how it does. I would appreciate any hints.
Hint: writing $U=\displaystyle\bigoplus_{i=1}^n U_i$ as a sum of $1$-dimensional $\Bbb C$-subspaces, there is an injection
$$\hom_G\left(\left(\bigoplus_{i=1}^n U_i\right)\otimes V^*,W\right)\to\bigoplus_{i=1}^n \hom_G\big(U_i\otimes V^*,W\big)$$
(Thinks of linear maps $U\otimes V^*\to W$ as bilinear maps $U\times V^*\to W$.)