Bornological space

Remarks

Equivalently, $E$ is bornological if and only if every linear map from $E$ to a locally convex space that maps bounded sets to bounded sets is continuous. Example: every normed space $E$ is bornological. If a convex balanced set $A$ absorbs the closed unit ball $B$, then $B \subseteq c A$ for some $c > 0$, so $A$ contains the ball of radius $1 / c$. More generally, every metrizable locally convex space is bornological. Non-example: let $E$ be an infinite-dimensional Banach space with its weak topology $\sigma(E, E')$. Its weakly bounded sets are norm bounded, by the uniform boundedness principle, so the closed unit ball absorbs all of them. But the ball is not a weak neighbourhood of $0$, since every weak neighbourhood of $0$ contains a nonzero linear subspace.