Bounded sets in topological vector spaces (Q1906877)

We study the relationship between the collection of bounded subsets of a metrizable topological vector space and various orderings on \(\mathbb{N}^{\mathbb{N}}\) related to the ordering of eventual dominance. This leads to a characterization of normability in metrizable topological vector sapces of large density in terms of the density of their bounded sets, and to a useful sufficient condition for a subspace of \(\mathbb{R}^{\mathbb{N}}\) to be barrelled. We also consider the problem of how sharp these results are by constructing several examples of spaces with small bounded subsets.