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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!!

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

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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

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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.

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Please see the following reference:

  1. Treves' book (Dover edition), page 450, exercise 44.6.

  2. this short article by Schwartz: http://www.numdam.org/item/SLS_1953-1954__1__A11_0

  3. and this long version of above http://sites.mathdoc.fr/OCLS/pdf/OCLS_1954-1955__25__88_0.pdf

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