Exercise 13, § 1

Let I be an infinite set, and $(E_i)_{i \in I}$ a family of locally convex spaces, none of which is 0. Let f be a linear mapping from $E = \prod_{i \in I} E_i$ into a Banach space F. Show that if the image of every bounded subset of E under f is a bounded subset of F, then there exists a finite subset H of I such that for every $i \notin H$, the restriction of f to $E_i$ (considered as a subspace of E) is null. (Argue by contradiction that if not, we can construct a bounded sequence $(x_n)$ in E whose image under f is unbounded in F.)