Proposition 3

— Let $(F_i)_{i \in I}$ be a family of barrelled spaces, and for every $i \in I$, let $f_i$ be a linear mapping from $F_i$ into a vector space E. The space E with the finest locally convex topology for which the $f_i$ are continuous (II, p. 27, prop. 5), is a barrelled space.

Let T be a barrel in E. Since $f_i$ is continuous, $f_i^{-1}(T)$ is a convex, balanced, closed and absorbent set in $F_i$; in other words, a barrel in $F_i$; since $F_i$ is barrelled, $f_i^{-1}(T)$ is a neighbourhood of 0 in $F_i$, for all $i \in I$. This implies that T is a neighbourhood of 0 in E (II, p. 27, prop. 5).