The approximation property for spaces of holomorphic functions on infinite-dimensional spaces I - Dineen, Seán; Mujica, Jorge (Journal of approximation theory, 2004)
For an open subset U of a locally convex space E, let ( H( U), τ 0) denote the vector space of all holomorphic functions on U, with the compact-open topology. If E is a separable Fréchet space with the bounded approximation property, or if E is a (DFC)-space with the approximation property, we show that ( H( U), τ 0) has the approximation property for every open subset U of E. These theorems extend classical results of Aron and Schottenloher. As applications of these approximation theorems we characterize the spectra of certain topological algebras of holomorphic mappings with values in a Banach algebra.