Proposition 11.4.4.label Let $\seqi{E}$ be bornological spaces over $K \in \RC$, $E$ be a vector space over $K$, and $\seqi{T}$ such that $T_{i} \in \hom(E_{i}; E)$ for all $i \in I$, then $E$ equipped with the inductive topology is bornolgic.
Proof. Let $\rho: E \to [0, \infty)$ be a seminorm on $E$ that is bounded on all bounded sets. For each $i \in I$ and $B \subset E_{i}$ bounded, $T_{i}(B)$ is bounded by Proposition 11.4.2, and $\rho \circ T_{i}(B)$ is bounded by assumption. Thus for every $i \in I$, $\rho \circ T_{i}$ is continuous, so $\rho$ is continuous by (4) of Definition 11.7.1.$\square$