Norm maximum principle for a Banach-space-valued holomorphic function ($\Vert f(z)\Vert \le {\sup }_{w\in \partial K}\Vert f(w)\Vert$)

For a compact set $K$ in the domain, a norming bounded linear functional at $f(z)$ reduces the assertion to the scalar maximum modulus principle. If $z$ lies inside $K$, use its connected component of the interior; that component's boundary lies in $\partial K$. If $z\in \partial K$, the inequality is immediate. No smoothness or connectedness assumption on $K$ is required.