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A weak criterion for vector-valued holomorphy. (English) Zbl 1055.46026

The author presents a new approach to a theorem, originally proved in K.–G. Große–Erdmann [“The Borel-Okada theorem revisited” (Univ. Hagen, FB Math. (Habil.)) (1992; Zbl 0953.30500)], which is a generalization of Dunford’s theorem (weak holomorphy implies holomorphy for Banach space valued functions). Here the image space is a locally complete space $E$. The main tool in the proof is J. Wolff’s theorem [C. R. Acad. Sci. Paris 173, 1327–1328 (1921; JFM 48.0320.01)].

It follows that a function $f$ defined on a subdomain ${\Omega }\setminus K$ of ${\Omega }\subseteq ℂ$ admits a unique holomorphic continuation to ${\Omega }$ if $f$ is locally bounded and the functions ${x}^{\text{'}}\circ f$ admit a holomorphic continuation, where ${x}^{\text{'}}$ runs through a separating subset of ${E}^{\text{'}}$, roughly speaking. This is also true under similar assumptions if $E$ is a Fréchet space.

In what follows, the author provides generalizations of his theorem to meromorphic functions and to functions defined on a domain of a locally convex spaces. Two results concerning a problem of Wrobel and a remark concerning real analytic functions conclude the paper.

##### MSC:
 46G20 Infinite dimensional holomorphy 30B40 Analytic continuation (one complex variable)
##### Keywords:
weak holomorphy; locally complete space