Let 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 denote the vector space of all complex-valued holomorphic germs on a compact subset K of E, with its natural inductive limit topology. Let denote the Banach space of all continuous complex-valued m-homogeneous polynomials on E. When E has a Schauder basis, we show that has the approximation property for every compact subset K of E if and only if has the approximation property for every . When E has an unconditional Schauder basis, we show that has the approximation property for every pseudoconvex open subset U of E if and only if has the approximation property for every . These theorems apply in particular to the classical Banach spaces and , and to the original Tsirelson space .1062457469