Exercise 2, § 2

Let I be an infinite set and $(E_i)_{i \in I}$ a family of locally convex spaces, none of which is 0. a) Suppose that each space $E_i$ is bornological. Show that if, in addition, the product space $\mathbf{R}^I$ is bornological, then the product $E = \prod_{i \in I} E_i$ is bornological (using III, p. 11, prop. 1 (iii)) and p. 39, exerc. 15, reduce to proving the following : a linear mapping $f$ from $E$ into a Banach space, which transforms every bounded set into a bounded set, and whose restriction to each $E_i$ is null, is necessarily null on $E$. For this consider, for every $x = (x_i) \in E$, the restriction of $f$ to the product of lines $\mathbf{R} x_i$. b) Deduce from a) that every product of a sequence ($E_n$) of bornological spaces is bornological.