Extension of entire functions on nuclear locally convex spaces

Abstract:

We prove that for a nuclear locally convex complex vector space every entire function is the pull-back of some entire function on an appropriate Banach space if and only if the entire functions on $E$ have the following universal extension property: whenever $E$ is a topological linear subspace of a locally convex space $F$ with a fundamental system of seminorms induced by semi-inner products, then every entire function $f$ on $E$ can be extended to an entire function on $F$.