Example 3

Every strict inductive limit E of a sequence $(E_n)$ of Montel spaces (II, p. 33) such that $E_n$ is closed in $E_{n+1}$ for all n, is a Montel space; in fact, E is Hausdorff (II, p. 32, prop. 9 (i)), barrelled (III, p. 25, cor. 3) and every bounded subset of E is contained in one of the $E_n$ (III, p. 5, prop. 6) hence is relatively compact in $E_n$, and consequently also in E.