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 E the union of the images of $E_p$. Now prove that the strong dual $E'_b$ of E is exhaustible (III, p. 50, exerc. 1). (Let $(U_j^p)$ be a decreasing fundamental system of convex, closed and balanced neighbourhoods of 0 in $E_p$; consider the finite intersections of the polar sets $(U_j^p)^o$ in $E'$; use the fact that for every increasing sequence $(m_j)_{j \geq 1}$, the intersection $\bigcap_{j=1}^\infty (U_{m_j}^j)^o$ is the polar of a neighbourhood of 0 in E, and the fact that $E'_b$ is complete.)