Exercise 16, § 3

Let $E$ be an inductive limit space of a sequence $(E_p)$ of locally convex metrizable spaces, the canonical mappings $E_p \to E$ being injective and let $E$ be the union of the images of the $E_p$. Show that the strong dual $E'_b$ of $E$ is exhaustible (III, p. 49, exerc. 1). (Let $(U_j^p)$ be a decreasing fundamental system of closed convex and balanced neighbourhoods of $0$ in $E_p$; consider the finite intersections of the polar sets $(U_j^p)^{\circ}$ in $E'$; use the fact that for every increasing sequence $(m_j)_{j \geq 1}$ the intersection $\bigcap_{j=1}^\infty (U_{m_j}^j)^{\circ}$ is the polar of a neighbourhood of $0$ in $E$, and that $E'_b$ is complete.)