Let $u,v$ be two distribution in $\mathcal{D}'(X),\mathcal{D}'(X')$:
given tensor product $$D'(X)\times D'(X') \to D'(X\times X')\\(u,v)\mapsto u\otimes v$$
prove tensor product is a separately continuous bilinear form on $D'(X)\times D'(X') $ a similar question
The question here is what's topology on $D'(X)$ ,do we use weak topology?
I can only show sequential continuous result that is $u_n\to u$ in weak topology ,then $u_n\otimes v \to u\otimes v$ in weak topology.
How to show the continuous result does sequential continuous = continuous for $D'(X)$ with weak topology?