Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)Let H(U) denote the vector space of all complex-valued holomorphic functions on an open subset U of a Banach space E. Let tau(omega) ancl tau(delta) 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), tau(omega)) and (H(U), tau(delta)) 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 arc actually valid for Riemann domains. (C) 2010 Elsevier Inc. All rights reserved.2592545560Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)FAPESP [06/02378-7]