Consider $C^\infty(X)$, equipped with its Fr'echet structure, and $\mathcal D'(Y)$, equipped with the strong topology, where $X,Y$ are compact manifolds. Is the strong dual of $C^\infty(X)\hat\otimes \mathcal D'(Y)$ a complete topological vector space? I have search all over the book of Treves for any hints but still am failing to determine it.
This dual is the same as $B(C^\infty(X),\mathcal D'(Y))$, the space of continuous bilinear maps $C^\infty(X)\times\mathcal D'(Y)\to \mathbb C$ (equipped with the product topology). However, I do not find any content on the regularity of this space, neither for general vector spaces nor for this specific example.
