Metrizability of spaces of holomorphic functions with the Nachbin topology (Q996900)

The authors show that the space of all holomorphic functions on an open subset \(U\) of a complex metrizable locally convex space \(E\), endowed with the Nachbin ported topology \(\tau_\omega\), is metrizable only if \(E\) has finite dimension. This answers a question posed in [\textit{J.\,Mujica}, ``Gérmenes holomorfos y funciones holomorfas en espacios de Fréchet'' (Publicaciones del Departamento de Teoría de Funciones, Universidad de Santiago, Spain) (1978; per bibl.)]. The following proposition is an important step of the proof and is of independent relevance. Let \(E\) be a metrizable locally convex space and let \(\tau\) be a locally convex topology between the compact open topology \(\tau_0\) and \(\tau_\omega\). If \((E',\tau)\) is metrizable, then \(E\) is a normed space.