Let $X=\operatorname{proj}\_nX\_n$ be a Fréchet space. What is the relation between $\operatorname{proj}\_nX\_n''$ and $X''$? This should be known but I cannot find the reference.



Projective description of a bidual of a Fréchet space
There is of course a canonical map $X''\to$ proj $X\_n''$ which however need not be surjective if $X$ is a *non-distinguished* Fréchet space because then the inductive limit topology on $X'=$ ind $X\_n'$ is strictly finer than the strong topology on $X'$. Examples are due to Köthe and Grothendieck and as far as I remember you should find one in the last chapter of Köthe's book. 


