Abstract
Abstract Let H (E) be the space of complex valued holomorphic functions on a complex Banach space E. The approximation property for H (E), endowed with various natural locally convex topologies, is studied. For example, H (E) with the compact-open topology has the approximation property if and only if E has the approximation property. In order to characterize when H (E) has the approximation property for topologies other than the compact-open, the notion of a compact holomorphic map between Banach spaces in introduced and studied.
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