I'm studying the book Uniform Algebras from Gamelin and I have some doubts about the proof of Theorem 1.1.
Theorem: Let $K$ be a compact subset of $\mathbb C$. If $f \in C(K)$ extends to be continuously differentiable in a neighborhood of $K$, and if $\partial f / \partial \overline z = 0$ on $K$, then $f \in R(K)$. Where $R(K)$ is the algebra of all functions in $C(K)$ which can be approximated uniformly on $K$ by rational functions with poles off $K$.
Proof: We can extend $f$ to be continuously differentiable on the complex plane, and to have compact support.
By Cauchy-Green's formula, $$ f(z) = \frac{-1}{\pi} \int \int \frac{\partial f}{\partial \overline \zeta} \frac{1}{\zeta - z}\, dx\, dy $$
where the integral is extended over $\mathbb C$. Let $\mu$ be a measure on $K$ which is orthogonal to $R(K)$.
By Fubini's Formula:
$$ \int f d\mu = \frac{-1}{\pi} \int \int \frac{\partial f}{\partial \overline \zeta}\, \left [ \int_K \frac{1}{\zeta - z}\, d\mu(\zeta) \right ]\, dx\, dy $$
When $z \in \mathbb C \setminus K$, the inner integer is zero. When $z \in K, \, \partial f / \partial \overline z = 0$. Then, $\int fd\mu = 0$.
Hence every continuous linear functional on $C(K)$ which is orthogonal to $R(K)$ is also orthogonal to $f$.
How can I proof that such an element $g$ in the dual $C(K)'$ is orthogonal to $f \in C(K)$?
And with this, how can I use the Hahn-Banach theorem to conclude that $f \in R(K)$?
Help?