Banach-space-valued holomorphic function ($F^{\prime }(z)={\lim }_{h\to 0}\frac{F(z+h)-F(z)}{h}$)

A map from an open subset of $\mathbb{C}$ to a complex Banach space is holomorphic when the displayed limit exists in the Banach-space norm at every point. A norm-convergent power series is holomorphic inside its radius of convergence: on every smaller disk the series of derivatives converges uniformly by the geometric coefficient bound, allowing differentiation term by term. Applying a continuous linear functional produces an ordinary scalar holomorphic function. Scalar Cauchy identities therefore imply the corresponding vector-integral identities when bounded linear functionals separate points, as they do by the Hahn-Banach theorem.

For a Banach algebra-valued holomorphic function, set $p_n(w)=\Vert f(w{)}^{2^n}{\Vert }^{1/2^n}$. These continuous functions decrease to the spectral radius by the spectral radius formula. If $M={\sup }_{\partial K}r(f(w))$, the continuous functions $\max \{p_n,M\}$ on $\partial K$ decrease to the continuous constant $M$. The Dini theorem gives uniform convergence. Apply the norm maximum principle for a Banach-space-valued holomorphic function to each power, and then let $n\to \infty$. This proof does not assume continuity of spectral radius.

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.