From Uniform Boundedness to the Boundary Between Convergence and Divergence

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