Irreducible characters form orthonormal basis of set of class functions

Question

I am reading Serre's book (Linear Representations of Finite Groups). Theorem 6 in chapter 2 says that the irreducible characters $\chi_1,\dotsc,\chi_h$ of a finite group $G$ form an orthonormal basis of $H$, the set of class functions on $G$. It says in the proof (given the $\chi_i$ form are orthonormal) that 'it is enough to show that every element of $H$ orthogonal to the $\chi_i^\ast$ is zero', where $\ast$ is the complex conjugate. Why is this so? Will appreciate any hints etc.

Answer 1

If $V$ is a finite dimensional inner product space, and $W$ is a subspace, then:

$$V=W\oplus W^{\perp}$$

Now, let $V$ be the space of class functions, and let $W$ be the subspace generated by the irreducible characters. Then the statement you've written means to show that $W^{\perp}=0$ from which it follows that $V=W$.

Perhaps we should also note that the subspace generated by irreducible characters is the same as that generated by their conjugates. Can you see why this is true?