Question

Let Fk(G) be the k-space of class functions on G, given the inner product
[μ, ν] = 1/|G| *(Sum over g) μ(g^−1)ν(g).
Show that, for any class function f ∈ Fk(G), there is a “Fourier expansion” f = (sum over i) [f,χi] χi.