Linear operators on normed 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 $E$ is bornological if every circled, convex subset $A \subset E$ that absorbs every bounded set in $E$ is a neighbourhood of $0$ in $E$. 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 $E$ with initial topology $T_0$, there is a finest topology $T$ such that the family of bounded subsets of $T$ coincides with $T_0$. The space $E$ equipped with the topology $T$ is called the bornologification of $E$, or the bornological space associated with $(E, T_0)$
Let $U$ be a linear map from a bornological space $E$ to any locally convex TVS, then the following statements are equivalent:
$U$ is continuous,
$U$ is bounded on bounded sets,
$U$ maps null sequences to null sequences.
Every inductive limit of Banach spaces is a bornological vector space. (Alpay-Salomon 13, prop. 2.3)
Conversely, every bornological vector space is an inductive limit of normed spaces, and of Banach spaces if it is quasi-complete (Schaefer-Wolff 99)
Every metrizible locally convex space is bornological, that is every Fréchet space and thus every Banach space.
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