Filter-dependent versions of the Uniform Boundedness Principle

Abstract

For every filter $\mathcal F$ on $\mathbb N$, we introduce and study corresponding uniform $\mathcal F$-boundedness principles for locally convex topological vector spaces. These principles generalise the classical uniform boundedness principles for sequences of continuous linear maps by coinciding with these principles when the filter $\mathcal F$ equals the Fr\'echet filter of cofinite subsets of $\mathbb N$. We determine combinatorial properties for the filter $\mathcal F$ which ensure that these uniform $\mathcal F$-boundedness principles hold for every Fr\'echet space. Furthermore, for several types of Fr\'echet spaces, we also isolate properties of $\mathcal F$ that are necessary for the validity of these uniform $\mathcal F$-boundedness principles. For every infinite-dimensional Banach space $X$, we obtain in this way exact combinatorial characterisations of those filters $\mathcal F$ for which the corresponding uniform $\mathcal F$-boundedness principles hold true for $X$.