TITLE: Topology on the dual of a Frechet space
QUESTION [5 upvotes]: If $F$ is a Frechet space, is there any locally convex space topology on the dual
$F'$, such that for each local diffeomorphism $f$ from an open subset $U$ of $F$ to $F$,
the map $U \times F' \longrightarrow F'$ is smooth?

REPLY [4 votes]: This map is not smooth for every vector space topology on the dual space, see Remark I.3.9. in Neeb, K.-H. "Towards a Lie theory of locally convex groups" 2006 for an explicit counterexample. 
Thus the cotangent bundle does not carry a (natural) smooth structure for Fréchet manifolds and one has to specify what one understands under smooth differential forms more directly. The usual way is to require the chart representation of the differential $k$-form $\alpha$ to be smooth as as a map $\alpha: U \times F^k \to \mathbb{R}$ (see Definition I.4.1 in the above paper).