Filter-dependent versions of the uniform boundedness principle

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