Weakly uniformly continuous holomorphic functions and the approximation property

We study the approximation property For spaces of Frechet and Gateaux holomorphic functions which are weakly uniformly continuous on bounded sets. We show when U is a balanced open subset of a Baire or barrelled metrizable locally convex space, E, that the space of holomorphic functions which are weakly uniformly continuous on LIT-bounded sets has the approximation property if and only if the strong dual of E, Eb, has the approximation property. We also characterise the approximation property for these spaces of vector-valued holomorphic functions in terms of the tensor product of the corresponding space of scalar-valued holomorphic functions and the range space.