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.

Answer 1

We know that the irreducible characters $\{\chi i\}$ form an orthonormal basis for $Fk(G)$ under $[, ]$. If $f = \sum_{i} a_{i} \chi_{i}$, we see immediately that $[f, \chi_{i}] = [\sum_{i} a_{i} \chi_{i}, \chi_{i}] = a_{i}$.