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Vector-valued holomorphic functions revisited. (English) Zbl 0976.46030
Let $U\subseteq ℂ$ open, $E$ a Banach-space and $W\subseteq {E}^{\text{'}}$ a subset. It is shown that all $\sigma \left(E,W\right)$-holomorphic mappings $f:U\to E$ are holomorphic iff all $\sigma \left(E,W\right)$-bounded subsets of $E$ are bounded. If $f$ is in addition locally bounded, then it is enough to assume that $W$ is separating. A Tauberian convergence result is proved, namely if the boundary values of a bounded sequence of holomorphic mappings on the disc converge on a subset of the boundary of positive Lebesgue measure then they converge on the disc. More general versions are proved using the Nevanlinna norm.
Reviewer: A.Kriegl (Wien)

##### MSC:
 46G20 Infinite dimensional holomorphy