Mathematical Proceedings of the Cambridge Philosophical Society



A weak criterion for vector-valued holomorphy


K.-G. GROSSE-ERDMANN a1
a1 Fachbereich Mathematik, FernUniversität Hagen, 58084 Hagen, Germany. e-mail: kg.grosse-erdmann@fernuni-hagen.de

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Abstract

We study functions $f$ of one complex variable with values in a complete locally convex space $E$. In [13] we had shown that for $f$ to be holomorphic it suffices that it is locally bounded and that there is a separating subset $H$ of $E'$ such that $x'\circ f$ is holomorphic for each functional $x'\in H$. We give here a new and considerably shorter proof of this result, and we extend it to a weak criterion for holomorphic continuability.

(Received April 29 2002)
(Revised October 21 2002)