= Weakly holomorphic Banach-valued function
{title2=$\varphi\circ f\text{ holomorphic for every }\varphi\in X'$}
A map from an open subset of $\mathbb C$ into a complex <Banach space> is weakly holomorphic when every composition with a bounded complex-<linear functional> is a scalar <holomorphic function>. <Norm> holomorphy requires its difference quotients to converge in the Banach <norm>. Weak holomorphy is initially a condition on scalar observations; the <weak holomorphy implies norm holomorphy> argument recovers an actual vector derivative. A closed target subspace containing values on a sequence with an interior accumulation point contains every value on the connected domain: annihilate the subspace, apply the scalar <identity theorem>, and use <Hahn-Banach separation>.
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