Holomorphic functions taking values in a quotient of Fréchet-Schwartz spaces

Abstract

We define a space of holomorphic functions $O_{1}(U,E\mid F)$ on a domain of holomorphy $U$ of ${\Bbb C}^{n}$, taking their values in quotient bornological spaces $E\mid F$ as the kernel of a sheaf-morphism. We show that if $E$ is a Schwartz $b$-space and $F$ is a Fréchet-Schwartz $b$-space, then $O_{1}(U,E\mid F)$ and $O\left( U,E\right) \mid O\left( U,F\right)$ are naturally isomorphic.