Let $U$ be an open set in $\mathbb{R}^n$ (or more generally, a manifold) and let $V$ be a separated bornological vector space. Do we have $$C^\infty(U, V) \cong C^\infty(U) \,\hat{\otimes}\, V,$$ as bornological vector spaces, where $\hat{\otimes}$ denotes the completed bornological tensor product?
Here I did not specify which bornologies to take on $C^\infty(U, V)$ and $C^\infty(U)$; is there a canonical one that makes the above result true?
