Abstract
We study the approximation property for spaces of Fréchet and Gâteaux 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 U-bounded sets has the approximation property if and only if the strong dual of E, E′ b , 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.
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