I read the following in Thirring's Quantum Mathematical Physics, 2nd ed., p. 17.
The dual space of the continuous functions on a compact set, with the norm $\operatorname{sup}_{z\in K} |f(z)|$ consists of the (not necessarily positive) measures on $K$.
$K$ is a compact subset of the complex plane.
Where can I find a reference for the above statement, or how would one go about proving such the above statement? I saw the Wikipedia page on the Riesz--Markov--Kakutani theorem. In the first version, the space of all continuous compactly supported functions is considered, but the theorem relates positive linear functionals, not all bounded linear functionals, as we need here. In the second version, the space considered is the space of continuous functions on a locally compact Hausdorff space which vanish on infinity.