Proposition 9

The strong dual of a Montel space is a Montel space.

Let E be a Montel space and $E'_b$ its strong dual. Since E is reflexive, $E'_b$ is barrelled (IV, p. 15, th. 1). Since every bounded subset of E is relatively compact the strong topology on $E'$ coincides with the topology of compact convergence. Let B be a bounded subset of $E'_b$; it is bounded for the weak topology $\sigma(E', E)$, hence is equicontinuous because E is barrelled. Then Ascoli’s theorem (GT, X, § 2, No. 4, cor. and § 2, No. 5, cor. 1) implies that the closure of B for $\sigma(E', E)$ is compact for the topology of compact convergence; therefore B is relatively compact in $E'_b$.