Exercise 8, § 4

Let $(E_i)_{i \in I}$ be a family of infrabarrelled spaces, and for every $i \in I$, let $f_i$ be a linear mapping from $E_i$ into a vector space E. Show that the space E, with the finest locally convex topology for which the $f_i$ are continuous, is infrabarrelled. In particular, every quotient space of an infrabarrelled space is infrabarrelled; every topological direct sum of infrabarrelled spaces is infrabarrelled.