Let $X$ be a complete and barrelled (DF)-space. It is known that the bidual $X''$ is a complete (DF)-space. Does it follow that it is also barrelled? More generally, is the bidual of a barrelled space again barrelled?



Biduals of barrelled spaces
According to Grothendieck, a Frechet space $E$ is distinguished if its dual (endowed with the strong topology $\beta(E',E)$ of uniform convergence on all bounded subsets of $E$) is barrelled (or, by a theorem of him equivalently, bornological or ultrabornological). $E'$ is thus a barrelled (DF)-space. [Here](http://link.springer.com/article/10.1007%2FBF01246746) Bonet, Dierolf, and Fernandez prove that there are distinguished Frechet spaces $E$ with non-distinguished bidual. Hence $X=E'$ is a barrelled (DF)-space whose bidual is not not barrelled.


