Extension of holomorphic mappings from 𝐸 to 𝐸”

Moraes, Luiza A.

doi:10.1090/S0002-9939-1993-1139471-0

Extension of holomorphic mappings from $E$ to $E”$
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by Luiza A. Moraes

Proc. Amer. Math. Soc. 118 (1993), 455-461

DOI: https://doi.org/10.1090/S0002-9939-1993-1139471-0

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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 $f|B$ 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 $\tilde f|B$ 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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