Question:
Let $K\subset \mathbb{C}^N$ be compact and let $\mathcal{P}(K)$ denote the closure of $\{p|_K:\, p \in \mathbb{C}[z_1,\dots,z_N]\}$ with respect to $||\cdot||_K$. Show that for each non zero, linear and multiplicative $\phi: \mathcal{P} \rightarrow \mathbb{C}$ there is a unique $z_{\phi}$ in the polynomially convex hull, $$\hat{K}:= \{z \in \mathbb{C}^N: \, |p(z)| \leq ||p||_K \text{ for all } p \in \mathbb{C}[z_1,\dots,z_N]\}$$such that $\phi(f)=f(z_{\phi})$ for all $f \in \mathcal{P}(K)$. It is assumed that $\phi$ is continuous.
So, I believe I've shown that we can find a unique $z_{\phi}$ such that $\phi(f)=f(z_{\phi})$. Indeed, since $\phi$ is multiplicative and non zero, it must be that $\phi(1)=1$. We then define $\phi(z_j)=z_{\phi,\, j}$ for some complex numbers $z_{\phi,\, j}$ with $j=1,\dots, N$. Then expanding $f$ as a power series and using the continuity and linearity of $\phi$ yields $\phi(f)=f(z_{\phi})$. However, I'm struggling to show that this $z_{\phi}$ belongs to $ \hat{K}$. I've tried using that $\phi$ is continuous and therefore bounded but that doesn't seem to help.
Any help would be much appreciated!