From Uniform Boundedness to the Boundary Between Convergence and Divergence

摘要

Summary We introduce a dual of the uniform boundedness principle that does not require completeness and gives an indirect means for testing the boundedness of a set. The dual principle, although known to analysts and despite its applications in establishing results such as the Hellinger-Toeplitz theorem, is often missing from elementary treatments of functional analysis. We give an example showing a connection between the dual principle and a question in the spirit of du Bois-Reymond regarding the boundary between convergence and divergence for 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.