Definition 9.3.1 (Bornologic Space). Let $E$ be a locally convex space, then the following are equivalent:
For any $U \subset E$ convex and balanced, if $U$ absorbs every bounded set of $E$, then $U \in \cn_{E}(0)$.
For any seminorm $\rho: E \to [0, \infty)$ that is bounded on all bounded sets of $E$, $\rho$ is continuous.
If the above holds, then $E$ is a bornologic space.
Proof. (1) $\Rightarrow$ (2): Let $B \subset E$ be bounded, then there exists $R > 0$ such that $\rho(B) \subset [0, R)$. In which case,
By assumption, $\bracs{\rho < 1}\in \cn_{E}(0)$, so $\rho$ is continuous by Lemma 9.1.9.
(2) $\Rightarrow$ (1): Let $\rho$ be the gauge of $U$, then for any $B \subset E$ bounded, there exists $R > 0$ such that $B \subset RU$. In which case, $\rho(B) \subset [0, R]$.$\square$