Spaceability of special families of null sequences of holomorphic functions

TL;DR

The paper addresses the spaceability of two convergence-gap sets for sequences of holomorphic functions on a nonempty open $Ω$: pointwise convergence to 0 but not compact on $Ω$ ( ${\mathcal S}_p \setminus {\mathcal S}_{uc}$ ) and compact convergence to 0 but not uniform ( ${\mathcal S}_{uc} \setminus {\mathcal S}_u$ ). Building on prior lineability results, it constructs closed infinite-dimensional subspaces $M(f)\subset H(Ω)^N$ that sit entirely inside these gaps, via two schemes: a partition-based construction using disjoint open pieces $S_k$, and an Arakelian-approximation-based approach to force non-uniform convergence. The main contribution is to establish spaceability in each difference set, containing prescribed members, thus strengthening earlier results that only showed large linear structures. These results deepen the understanding of pathological convergence behavior in spaces of holomorphic function sequences and provide tools for assembling large closed algebraic structures within these sets.

Abstract

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

Theorems & Definitions (11)

• Theorem 2.1
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Theorem 3.3
• proof
• Lemma 3.4
• proof
• Theorem 3.5