The approximation property for spaces of holomorphic functions on infinite dimensional spaces III

Abstract

Let $(\mathcal{ H} (U), \tau _{\omega })$ denote the vector space of all complex-valued holomorphic functions on an open subset U of a Banach space E, with the Nachbin compact-ported topology. Let $(\mathcal{ H} (K), \tau _{\omega })$ denote the vector space of all complex-valued holomorphic germs on a compact subset K of E, with its natural inductive limit topology. Let $\mathcal{ P} (^{m}E)$ denote the Banach space of all continuous complex-valued m-homogeneous polynomials on E. When E has a Schauder basis, we show that $(\mathcal{ H} (K), \tau _{\omega })$ has the approximation property for every compact subset K of E if and only if $\mathcal{ P} (^{m}E)$ has the approximation property for every $m \in \mathbb{ N}$. When E has an unconditional Schauder basis, we show that $(\mathcal{ H} (U), \tau _{\omega })$ has the approximation property for every pseudoconvex open subset U of E if and only if $\mathcal{ P} (^{m}E)$ has the approximation property for every $m \in \mathbb{ N}$. These theorems apply in particular to the classical Banach spaces $\ell _{1}$ and $c_{0}$, and to the original Tsirelson space $T^{*}$.