Exercise 12, § 3

a) Let F be a separated (DF)-space, $F'_b$ its strong dual. Now prove that if $F'_b$ is reflexive, the completion $\hat{F}$ of F is reflexive and equal to the bidual $F''$ of F ($cf.$ IV, p. 52, exerc. 4 et p. 54, exerc. 11). b) Let E be a Fréchet space. Now prove that if the bidual $E''$ of E is reflexive, then E is reflexive.