Fréchet-valued symbols

Question

Denote by $S^m \left ( \mathbb{R}^k \times \mathbb{R}^n \right)$ the usual space of symbols. Now let $E$ be a Fréchet Space. We can then define $S^m \left ( \mathbb{R}^k \times \mathbb{R}^n; E \right)$ as the space of smooth functions $q:\mathbb{R}^k \times \mathbb{R}^n \to E $ which satisfy $ \sup_{z,\xi} || \partial_z^\alpha \left < \xi \right >^{|\beta|-m} \partial_\xi^\beta q ||_j < \infty$ for a family of seminorms $||\cdot||_j$ for $E$.

My question is: under which conditions can we say that $S^m \left ( \mathbb{R}^k \times \mathbb{R}^n; E \right)$ is isomorphic (via the natural map) to the completed projective tensor product $S^m \left ( \mathbb{R}^k \times \mathbb{R}^n \right) \hat \otimes E$?

I'm sure these things have been looked at before. Any suggestion on papers using them are welcome.

Answer 1

Check if this follows with appropriate adjustments from the arguments for Thm.44.1 and Exr.44.6 in

Trèves, François, Topological vector spaces, distributions and kernels, Pure and Applied Mathematics (Academic Press) 25. New York-London: Academic Press. XVI, 565 p. (1967). ZBL0171.10402.

The only condition there is that $E$ is a complete locally convex Hausdorff topological vector space and $\otimes$ stands for the completed injective tensor product. If you need to insist on the projective tensor product, you may have to invoke nuclearity.