While reading some papers about $\psi$DOs I found some spaces of vector valued functions which I am not familiar with. I am looking for references about the Schwartz space of functions with values in a Frechet space $E$ i.e. $\mathcal{S}(\mathbb{R},E)$. 


For example when $E=L^{m}\_{Cl}(\Omega)$ is the space of $\psi$DOs with classical symbols the space $\mathcal{S}(\mathbb{R},E)$ would be a space of 1-parameter families of $\psi$DOs with certain decay properties when you approach $+\infty$ and $-\infty$. 


When $E$ is a Banach space this space is well-known and there are plenty of references, however I have been not able to find references when $E$ is a Frechet space. I already had a look to Shaefer's book on Topological Vector Spaces and also I check Treves' book TVS Distributions and Kernels but there is nothing about $\mathcal{S}(\mathbb{R},E)$ or even the Fourier transformation on Frechet-valued functions. 


I would like to ask for references about Frechet valued Schwartz spaces.


Thanks!! 



Schwartz space of functions with values in a Frechet space
$\mathcal S(M,E) = \mathcal S(M)\bar\otimes E$ for the completed injective or projective tensor product, which agree since $\mathcal S(M)$ is a nuclear spaces. See H. Jarchow. Locally convex spaces. Teubner, Stuttgart, 1981. Fourier transform you can apply just to the left hand side of the tensor product.


Also, have a look at page 533 of the book of Treves you cited. This is treated there.


