Linear operators on Hilbert spaces are continuous precisely iff they are bounded. A bornological space retains this property by definition.
The discussion below is about bornological CVSes, but there is a more general notion of bornological space.
A locally convex topological vector space is bornological if every circled, convex subset that absorbs every bounded set in is a neighbourhood of in . Equivalently every seminorm that is bounded on bounded sets is continuous.
The bornology of a given TVS is the family of bounded subsets.
Given a locally convex TVS with initial topology , there is a finest topology such that the family of bounded subsets of coincides with . The space equipped with the topology is called the bornologification of , or the bornological space associated with
Let be a linear map from a bornological space to any locally convex TVS, then the following statements are equivalent:
is bounded on bounded sets,
maps null sequences to null sequences.
Wikipedia about bornological spaces
H. H. Schaefer with M. P. Wolff, section 8 of Topological vector spaces, Springer 1999
Daniel Alpay, Guy Salomon, On algebras which are inductive limits of Banach spaces (arXiv:1302.3372)
Discussion of bornological vector spaces forming a quasi-abelian category is in
with review and generalization to bornological abelian groups in