What Serre does is take a finite group $G$ with irreducible characters $\chi_1,\dots,\chi_n$ and assumes $f$ is in the orthogonal complement of the $\chi_1^{\ast},\dots,\chi_n^{\ast}$. For a rep $\rho$ on $V$, he then looks at the endomorphism $\sum_{g\in G}f(g)\rho(g)$ of $V$. This is seen to be equivariant, whence a homothety by Schur's lemma and its ratio is computed as $\frac{|G|}{\dim(v)}\langle f,\chi_{\rho}^{\ast}\rangle$. This vanishes for all irreps by hypothesis, whence for all reps by additivity. You then apply this to the regular rep and obtain $0=\sum_{g\in G}f(g)g$, whence $f\…