Let $E \to M$ and $F \to N$ be two vector bundles over smooth manifolds $M, N$. Denote by $\pi_1, \pi_2$ the projections of $M \times N$ to $M$ and $N$, respectively. Equip the spaces of sections of a vector bundle with its usual Fréchet space structure to make it into locally convex vector spaces.
If we denote by $E \boxtimes F := \pi_1^* E \otimes \pi_2^* F$ the external tensor product vector bundle over $M \times N$, is there then a natural vector space isomorphism $$ \Gamma(E \boxtimes F)^* \cong \left( \Gamma(E) \otimes_{\mathbb{R}} \Gamma(F) \right)^*?$$
The dual on the left-hand side denotes the continuous dual of a locally convex vector space, the dual on the right-hand side denotes the space of jointly continuous, $\mathbb{R}$-bilinear maps.
I think this is folklore, but I cannot for the life of me find a source proving this precise result. Is there a canonical source to go to? Is this somewhere in Dieudonné's "Eléments d'analyse"?