= Banach-space-valued holomorphic function
{c}
{title2=$F'(z)=\lim_{h\to0}\frac{F(z+h)-F(z)}h$}
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>.
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