Definition:Analytic Function/Banach Space Valued 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 $f : U \to X$ be a mapping.

We say that $f$ is analytic if and only if the limit:

$\ds \lim_{w \mathop \to z} \frac {\map f w - \map f z} {w - z}$

exists for each $z \in U$.