Let $X,Y$ be smooth compact manifolds. Let $C^\infty(X)$ and $C^{-\infty}(X)$ denote the spaces of smooth and generalized functions on $X$ respectively. We have the obvious canonical linear map $$T\colon C^{-\infty}(X\times Y)\to Bil(C^\infty(X),C^\infty(Y)),$$ where the target is the space of continuous bilinear functionals $C^\infty(X)\times C^\infty(Y)\to \mathbb{C}$. By definition $(T\Phi)(f,g)=\Phi(f\otimes g)$.

It is well known that $T$ is isomorphism of vector spaces. **For which standard topologies on the source and the target $T$ is an isomorphism of topological vector spaces?**

For example let us choose on $C^{-\infty}(X\times Y)$ the strong topology, and on the target the topology given by seminorms
$$||B||_{K,L}:=\sup_{k\in K,l\in L}|B(k,l))|,$$
where $K\subset C^\infty(X), L\subset C^\infty(Y)$ are arbitrary bounded subset. Then $T$ will be a continuous map for these topologies. **But is $T$ a topological isomoprhism?**