Filter-dependent versions of the uniform boundedness principle

Abstract

For every filter F on N, we introduce and study corresponding uniformF-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 F equals the Fréchet filter of cofinite subsets of N. We determine combinatorial properties for the filter F which ensure that these uniform F-boundedness principles hold for every Fréchet space. Furthermore, for several types of Fréchet spaces, we also isolate properties of F that are necessary for the validity of these uniform F-boundedness principles. For every infinite-dimensional Banach space X, we obtain in this way exact combinatorial characterisations of those filters F for which the corresponding uniform F-boundedness principles hold true for X.