DOI link for Bornological Spaces
Bornological Spaces book
Continuous linear maps are always locally bounded in the sense that they map bounded sets into bounded sets [Theorem 6.4.1]. If the domain is pseudometrizable, the converse holds: Local boundedness implies continuity [Theorem 6.5.2]. The purpose of this chapter is to identify the class of LCS X for which all locally bounded linear maps into any LCS Y are continuous. Mackey  first singled out this class of spaces, which we call bornological, the term coined by Bourbaki.