Spaceability of special families of null sequences of holomorphic functions

TL;DR

This paper demonstrates the existence of large, closed, infinite-dimensional subspaces within sequences of holomorphic functions, where sequences exhibit specific convergence behaviors, expanding understanding of the structure of such function spaces.

Contribution

It establishes the spaceability of sequences of holomorphic functions with particular pointwise and compact convergence properties, complementing previous research.

Findings

Existence of two closed infinite-dimensional subspaces with specific convergence properties.

Sequences tending to zero pointwise but not compactly.

Sequences tending to zero compactly but not uniformly.

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.