Definition 5

Let $E$ be a locally convex space. The set of all bounded subsets of $E$ is called the canonical bornology of $E$.

If $\mathscr{B}$ is a set of bounded subsets of $E$, then there exists a smallest bornology $\tilde{\mathscr{B}}$ adapted to $E$ and containing $\mathscr{B}$. The sets of $\tilde{\mathscr{B}}$ are those that are contained in a set homothetic to the closed convex balanced envelope of a finite union of sets of $\mathscr{B}$.

Every adapted bornology is contained in the canonical bornology.