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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.

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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.

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I assume that you mean one of the essentially equivalent representatios of the space as a projective limit of a sequence of Banach spaces. Your object of enquiry is then not in general the bidual in the sense of lcs's but the dual of $X'$ provided with its natural bornology, i.e., it is the space of linear functionals on $X'$ which are bounded on the equicontinuous subsets. This can be larger than the lcs bidual though it coincides with it in many cases. At the moment I have no access to the literature and so cannot give precise references but would refer you to Grothendieck on $DF$-spaces (in particular, his book on lcs's) and the monographs of Hogbe-Nlend on convex bornological spaces.

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