Metrizability of spaces of holomorphic functions

In this paper we prove that if U is an open subset of a metrizable locally convex space E of infinite dimension, the space H ( U ) of all holomorphic functions on U, endowed with the Nachbin–Coeuré topology τ δ , is not metrizable. Our result can be applied to get that, for all usual topologies, H ( U ) is metrizable if and only if E has finite dimension.