Bornological space

Summary

A bornological space is a locally convex space whose topology is determined by its bounded sets. In such a space, every convex balanced set that absorbs all bounded sets is a neighbourhood of $0$. The main consequence concerns a linear map from a bornological space to a locally convex space. Such a map is continuous as soon as it maps bounded sets to bounded sets. Every metrizable locally convex space, in particular every normed space, is bornological.