Definition:Weakly Analytic Function

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Let $U$ be an open subset of $\C$.

Let $\struct {X, \norm {\, \cdot \,} }$ be a Banach space over $\C$.

Let $\struct {X^\ast, \norm {\, \cdot \,}_{X^\ast} }$ be the normed dual space of $\struct {X, \norm {\, \cdot \,} }$.

Let $f : U \to X$ be a mapping.

We say that $f$ is weakly analytic if and only if:

for each $\phi \in X^\ast$, $\phi \circ f : U \to \C$ is analytic.

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