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Extension of holomorphic mappings from $ E$ to $ E''$

Author: Luiza A. Moraes
Journal: Proc. Amer. Math. Soc. 118 (1993), 455-461
MSC: Primary 46G20
MathSciNet review: 1139471
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Abstract: Assuming that $ E$ is a distinguished locally convex space and $ F$ is a complete locally convex space, we prove that there exists an open subset $ V$ of $ E''$ that contains $ E$ and such that every holomorphic mapping $ f:E \to F$ whose restriction $ f\vert B$ is $ \sigma (E,E')$-uniformly continuous for every bounded subset $ B$ of $ E$ has a unique holomorphic extension $ \tilde f:V \to F$ such that $ \tilde f\vert B$ is $ \sigma (E'',E')$-uniformly continuous for every bounded subset $ B$ of $ V$. We show that in many cases we can take $ V = E''$. This is the case when $ E''$ is a locally convex space where every $ G$-holomorphic mapping that is bounded in a neighbourhood of the origin is locally bounded.

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