Modes of convergence of sequences of holomorphic functions: a linear point of view

TL;DR

This paper compares different modes of convergence of holomorphic function sequences using linear algebra, revealing large algebraic structures of sequences with specific convergence behaviors.

Contribution

It demonstrates the existence of large linear algebras and dense subspaces of holomorphic sequences with various convergence properties, from a linear perspective.

Findings

Existence of large linear algebras of sequences tending to zero in different modes.

Construction of dense linear subspaces in Fréchet spaces.

Identification of infinite-dimensional Banach spaces of converging sequences.

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 Fr\'echet space as well as infinite dimensional Banach spaces of sequences converging to zero in the mentioned modes are shown to exist.