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

Abstract

Let H ( U ) denote the vector space of all complex-valued holomorphic functions on an open subset U of a Banach space E. Let τ ω and τ δ respectively denote the compact-ported topology and the bornological topology on H ( U ) . We show that if E is a Banach space with a shrinking Schauder basis, and with the property that every continuous polynomial on E is weakly continuous on bounded sets, then ( H ( U ) , τ ω ) and ( H ( U ) , τ δ ) have the approximation property for every open subset U of E. The classical space c 0 , the original Tsirelson space T ∗ and the Tsirelson ∗–James space T J ∗ are examples of Banach spaces which satisfy the hypotheses of our main result. Our results are actually valid for Riemann domains.