Metrizability of spaces of holomorphic functions with the Nachbin topology

In this paper we prove, among other things, that the space of all holomorphic functions on an open subset U of a complex metrizable space E, endowed with the Nachbin ported topology, is metrizable only if E has finite dimension. This answers a question by Mujica in [J. Mujica, Gérmenes holomorfos y funciones holomorfas en espacios de Fréchet, Publicaciones del Departamento de Teoría de Funciones, Universidad de Santiago, Spain, 1978].