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 PI:E --> E/F be the canonical quotient mapping. In this paper we study the induced mapping PI*, taking f is-an-element-of H(b)(PI(U))--> f circle PI is-an-element-of H(b)(U), where H(b)(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 Frechet-Schwartz space, and that it is not an embedding for certain subspaces F of every Frechet-Montel, not Schwartz, space. We provide several examples in the case where E is a Banach space to illustrate the sharpness of our results.