# Schwartz space of functions with values in a Frechet space

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!!

$\mathcal S(M,E) = \mathcal S(M)\bar\otimes E$ 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.

As an addendum to the above response, you might be interested in the fact that Schwartz wrote a sequel to his classic under the title "Théorie des distributions vectorielles" which gives an exhaustive treatment of this theme. It appeared in the Fourier Annals and is easily available online

Of course, the ultimate reference on this kind of stuff is the sequel mentioned by corserine, namely, this paper and this other one by Schwartz. I did not get a chance to look at the book from Peter's answer, but another useful reference (available online!) is "Vector-Valued Distributions And Fourier Multipliers" by Herbert Amann.