Example

— 1) Every semi-normed space is bornological. 2) In particular, every finite dimensional locally convex space is bornological. 3) On account of the transitivity of final locally convex topologies (II, p. 28, cor. 2), we deduce at once from condition (ii') that if (E_i)_{i \in I} is a family of locally convex bornological spaces and if E is assigned the finest locally convex topology for which the linear mappings u_i : E_i \to E (for i \in I) are continuous, then E is bornological. In particular, an inductive limit, a direct sum, a quotient space of bornological spaces are bornological spaces.

On the other hand, a closed subspace of a bornological space is not necessarily bornological (IV, p. 63, exerc. 8).