Aron--Berner--type extension in complex Banach manifolds

László Lempert

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arXiv · 2303.12047

Aron--Berner--type extension in complex Banach manifolds

László Lempert

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Abstract

Let $S$ be a compact Hausdorff space and $X$ a complex manifold. We consider the space $C(S,X)$ of continuous maps $S\to X$, and prove that any bounded holomorphic function on this space can be continued to a holomorphic function, possibly multivalued, on a larger space $B(S,X)$ of Borel maps. As an application we prove two theorems about bounded holomorphic functions on $C(S,X)$, one reminiscent of the Monodromy Theorem, the other of Liouville's Theorem.

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László Lempert. 2023-03-21. Aron--Berner--type extension in complex Banach manifolds. https://arxiv.org/abs/2303.12047

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