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 $E$ 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 metrizible locally convex space is bornological, that is every Fréchet space and thus every Banach space.
Wikipedia about bornological spaces
Helmut Schäfer: Topological Vector Spaces, chapter 8.