I am reading a proof of the Stone-Weierstrass theorem by De Branges, but I am having trouble understanding the following part.
Let $E$ be a locally compact Hausdorff space and $C_0(E, \mathbb{C})$ the continuous functions vanishing at infinity. For a vector subspace $S \subset C_0(E, \mathbb{C})$, define the set $M(S)$ of all signed measures $\mu$ on $E$ satisfying the following two properties: $\mu$ has a total variation of at most 1 and for all $f \in S$ we have $\int f \, d\mu = 0$. If $M(S)$ is equipped with the weak-*topology, then it is compact.
I have tried the following. By the Riesz-Markov-Kakutani representation theorem, the measures in $M(S)$ correspond to functionals in the dual space. I tried to show that the corresponding functionals are contained in a closed ball of any radius centered at the origin, since then the Banach-Alaoglu theorem would imply compactness. However, I believe boundedness of the functions $f \in C_0(E, \mathbb{C})$ is required, which I cannot show.
Any hints would be helpfull. Am I proceeding correctly? Thanks :)