Spaceability of special families of null sequences of holomorphic functions

Abstract

In this note, we consider the space $H(\Omega {)}^{\mathbb{N}}$ of sequences of holomorphic functions on an open set $\Omega \subset \mathbb{C}$. If $H(\Omega )$ is endowed with its natural topology and $H(\Omega {)}^{\mathbb{N}}$ is endowed with the product topology, then it is proved the existence of two closed infinite dimensional vector subspaces of $H(\Omega {)}^{\mathbb{N}}$ such that all nonzero members of the first subspace are sequences tending to zero pointwisely but not compactly on $\Omega$ and all nonzero members of the second subspace are sequences tending to zero compactly but not uniformly on $\Omega$. This complements the results provided in a recent work by the same authors.