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To prove the existence of Fourier expansion, I have to solve the following exercise, which supposedly follows from the Stone-Weierstrass theorem:

Let $G$ be a compact abelian topological group with Haar measure $m$. Let $\hat G$ be the dual. The members of $\hat G$ form an orthonormal basis for $L^2(m)$.

Here I do not know about existence of Haar measure and I took it for granted. I was able to show that the members of $\hat G$ are orthonormal. How to show that these form a basis? Stone-Weierstrass(whose proof too I don't know), reads as:

Let $X$ be a compact Hausdorff space and let $A$ be a closed subalgebra of the space of complex continuous functions $\mathcal C(X,\mathbb C)$ which separates points, contains a nonzero constant function and contains the conjugate of each of its functions. Then $A$ equals $\mathcal C(X,\mathbb C)$.

Here it is easy to show everything except the fact that the subspace of $L^2(m)$ generated by the characters separate points. How to do this?

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    I don't understand the question. Do you mean that the characters form an orthonormal basis for $C(G,\mathbb C)$?2011-09-20
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    @Stefan: Sorry, fixed.2011-09-20

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