Answer to Question #18552 in Abstract Algebra for Mohammad

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 two class functions f, f' ∈ Fk(G), there is a “Plancherel formula”
[f, f'] = (sum over i) [f,χi] [f', χi].