Exercise 6, § 1

Let E be a locally convex space, which is the strict inductive limit of a strictly increasing sequence $(E_n)$ of locally convex Hausdorff spaces, each $E_n$ being closed in $E_{n+1}$ (II, p. 32, prop. 9). a) Prove that E is not metrizable (use III, p. 5, prop. 6 and the preceding exerc. 5). b) In order that there exist a countable base for the canonical bornology of E, it is necessary and sufficient that the canonical bornology of each $E_n$ has a countable base in $E_n$.