Exercise 1, § 2

Let E be a locally convex space, and $\mathscr{T}$ its topology. Amongst the locally convex topologies on E for which the bounded sets are the same as those for $\mathscr{T}$, there is one $\mathscr{T}'$ finer than all the others, and this is the only one amongst these topologies which is bornological. The space obtained by assigning E with $\mathscr{T}'$ is called the bornological space associated with E. A linear map $u$ from E into a locally convex space F transforms every bounded subset of E into a bounded subset of F if and only if it is continuous for the topology $\mathscr{T}'$. Show that the topology $\mathscr{T}'$ is the finest of the locally convex topologies on E for which the canonical injections $E_A \to E$, where A runs through the family of convex, bounded and balanced subsets of E, are continuous.