The dual of the space of holomorphic functions on localy closed convex sets

Abstract

Let $H(Q)$ be the space of all the functions which are holomorphic on an open neighbourhood of a convex locally closed subset~$Q$ of $\mathbb{C}^N$, endowed with its natural projective topology. We characterize when the topology of the weighted inductive limit of Frechet spaces which is obtained as the Laplace transform of the dual $H(Q)'$ of $H(Q)$ can be described by weighted sup-seminorms. The behaviour of the corresponding inductive limit of spaces of continuous functions is also investigated.