From Uniform Boundedness to the Boundary Between Convergence and Divergence

TL;DR

This paper introduces a dual version of the uniform boundedness principle that does not require completeness, providing a new way to test boundedness and exploring its connection to convergence-divergence boundaries.

Contribution

It presents a dual form of the uniform boundedness principle applicable without completeness, filling a gap in elementary functional analysis treatments.

Findings

Connects the dual principle to convergence-divergence boundary questions

Clarifies the naturalness and scope of the dual principle

Illustrates the principle with an example related to sequence behavior

Abstract

In this article we introduce a dual of the uniform boundedness principle which does not require completeness and gives an indirect means for testing the boundedness of a set. The dual principle, although known to the analyst and despite its applications in establishing results such as Hellinger--Toeplitz theorem, is often missing from elementary treatments of functional analysis. In Example 1 we indicate a connection between the dual principle and a question in spirit of du Bois-Reymond regarding the boundary between convergence and divergence of sequences. This example is intended to illustrate why the statement of the principle is natural and clarify what the principle claims and what it does not.