Spaces of Holomorphic Functions and Germs on Quotients

Abstract

Let E be a complex locally convex space, F a closed subspace of E and let Π be the canonical quotient mapping from E onto E/F . If U is an open subset of E and K is a compact subset of E , Π induces a linear injective mapping Π * from [Π( U )] (resp. [Π( K )]) into ( U ) (resp. ( K ), where ( V ) (resp. ( L )) stands for the space of holomorphic functions (resp. germs) on the open subset V (resp. the compact subset L ) of a complex locally convex space. This mapping Π * is defined by Π * ( g ) = g ○ Π; g ∈ [Π( U )] (resp. [Π( K )]). When we consider the natural topologies on the spaces of holomorphic functions and holomorphic germs, Π * is always continuous. In some cases Π * is also a topological isomorphism onto its image (embedding) but this is not always true. The aim of this paper is to give a survey of the results concerning when Π * is, and is not, an embedding. We also include some new results with their proofs.