Definition:Analytic Function/Banach Space Valued Function
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Definition
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 $z \in U$.
Sources
- 2011: Graham R. Allan and H. Garth Dales: Introduction to Banach Spaces and Algebras ... (previous) ... (next): $3.2$: Integration of continuous vector-valued functions