Modes of convergence of sequences of holomorphic functions: a linear point of view
L. Bernal-González, M. C. Calderón-Moreno, J. López-Salazar, J. A. Prado-Bassas
TL;DR
The paper investigates how many sequences of holomorphic functions on an open subset of the complex plane converge to zero under one mode but not another. It uses a linear-analytic approach, combining Runge approximation, algebraic bases, and exponential multipliers to build large algebraic and Banach structures inside the convergence-gap sets $S_p - S_{uc}$ and $S_{uc} - S_u$, including free algebras and infinite-dimensional subspaces. Key contributions include proving strong $c$-algebrability of both gap sets, establishing pointwise continuum-lineability, and embedding infinite-dimensional Banach spaces inside these families, along with a discussion of spaceability and topological size via complete metrics and category arguments. The results illuminate rich interactions between convergence notions in complex analysis and linear structure, offering insights with potential implications for the study of holomorphic sequences and their asymptotic behavior.
Abstract
In this paper, pointwise convergence, uniform convergence and compact convergence of sequences of holomorphic functions on an open subset of the complex plane are compared from a linear point of view. In fact, it is proved the existence of large linear algebras consisting, except for zero, of sequences of holomorphic functions tending to zero compactly but not uniformly on the open set or of sequences of holomorphic functions tending pointwisely to zero but not compactly. Also dense linear subspaces in an appropriate Fréchet space as well as infinite dimensional Banach spaces of sequences converging to zero in the mentioned modes are shown to exist.