Embeddings of Spaces of Holomorphic Functions of Bounded Type

Abstract

Let U be an open subset of a complex locally convex space E, let F be a closed subspace of E, and let п:E/F be the canonical quotient mapping. In this paper we study the induced mapping п∗, taking f ɛ Hb{п(U)) → fo п ɛ Hb, where Hb(V) denotes the space of holomorphic functions of bounded type on an open set V. We prove that this mapping is an embedding when E is a Fréchet-Schwartz space, and that it is not an embedding for certain subspaces F of every Fréchet-Montel, not Schwartz, space. We provide several examples in the case where E is a Banach space to illustrate the sharpness of our results.