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!!