From Uniform Boundedness to the Boundary Between Convergence and Divergence

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.

MSC::

• Primary 40A05
• Secondary 46-01

Additional information

Notes on contributors

Ehssan Khanmohammadi

Ehssan Khanmohammadi (MR Author ID: 916345) received his Ph.D. in Mathematics from The Pennsylvania State University. His research interests include spectral theory, harmonic analysis, and mathematical physics.

Omid Khanmohamadi

Omid Khanmohamadi (MR Author ID: 861820) has earned a Master’s in Mechanical & Aerospace Engineering as well as Master’s and Ph.D. in Applied & Computational Mathematics. His research has spanned a wide range of topics, including numerical integration on manifolds, spectral methods for PDEs and inverse problems, numerical analysis of nonnormal operators, and high performance computational software design.