Spaces of Holomorphic Germs on Quotients

Let (X, Φ) be a Riemann domain over a complex Fréchet space E, K a compact subset of X and H (K) the vector space of all holomorphic germs on K. Given an F-quotient (XF, φF, ψ) of (X, φ) (see the definition below), the canonical mapping ψ : X ↦ XF induces a mapping ψ* : g ∈ H (ψ(K)) ↦ g ○ ψ ∈ H (K). Our aim here is to study conditions under which ψ* is an embedding when natural topologies on spaces of germs are considered.