2020/01/06 by De Bondt, Ben, Vernaeve, Hans
#03E20 #46B99 #FOS: Mathematics #Functional Analysis (math.FA) #Logic (math.LO) #Primary 46A04 #Secondary 40A35
paper · doi:10.48550/arxiv.2001.01663
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 Fréchet 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 Fréchet space. Furthermore, for several types of Fréchet 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.