Suppose we have a compact group $G$ with continuous $f$ on $G$ that is also $G$-finite.
I am told that then, out of all the irreducible representations $\pi$ of $G$ we must have $\pi(f)\neq 0$ only for finitely many. Here the representation is thought of as a transformation on the space of functions.
Why is this true? I feel like it is very short, and I was just missing something quick. Can someone enlighten me on this one.
(Also, by $G$-finite I mean that the vector space of right and left translates is finite.)