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 $$T^{*}$$ .