A remark about the embedding $(H(E/F),tau)to(H(E),tau)$, with $tau=tau₀,tau_omega$, in Frechet spaces

Abstract

In a recentpaper by Aron-Moraes-Ryan [2], it is proved that when E is a complex Banach space, F is a closed subspace of E and U is a balanced open subset of E, then the mapping $f ∈ H(?(U)) → f ○ ? ∈ H(U)$ where ? is the canonical mapping from E onto $E/F$, is a topological isomorphism from $(H(?(U)),τ)$ onto a closed subspace of $(H(U),τ$, where $τ = τ₀,τ_ω$. The aim of this remark is to show that the same result is true, with $τ₀$ for Fréchet spaces, and with $τ_omega$ for Fréchet-Schwartz spaces. Also we prove that this result is not true, with $τ_ω$ for some Fréchet-Montel spaces and with $τ_δ$ for some nuclear Fréchet spaces.