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I ran into the following definition of tame Frechet spaces and Nash-Moser therem.

It says that the space of smooth functions on a compact manifold is tame Frechet.

However, I wonder if

The Schwartz space $\mathcal{S}(\mathbb{R}^N)$ is also a tame Frechet space, even if $\mathbb{R}^N$ is not compact.

At least I was able to show and got confirmed that the Schwartz space is a graded Frechet space.

However, according to the definition of tame Frechet spaces stated in the top link, I have to find some Banach space $B$ and maps $L : \mathcal{S}(\mathbb{R}^N) \to \Sigma(B)$, $M: \Sigma(B) \to \mathcal{S}(\mathbb{R}^N)$ with required "boundedness" property.

I stronlgy suspect that $B = L^\infty(\mathbb{R}^N)$ with \begin{equation} L(f) := \bigl( e^{-e^{k-1}} f \bigr)_{k \in \mathbb{N}} \end{equation} must somehow work. However I have difficulty finding the appropriate $M : \Sigma(B) \to \mathcal{S}(\mathbb{R}^N)$.

Could anyone please help me fill in all the details? I have not copied every definition from the link tame Frechet spaces and Nash-Moser therem, but I will do so if anyone requests for the sake of completeness.

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    $\begingroup$ $\mathscr S$ is isomorphic to the space $s$ of rapidely decreasing sequences which itself is isomorphic to the space of smoooth functions on the circle $S^1$. You find these isomorphisms, e.g., in the book Introduction to Functional Analysis of Meise and Vogt. $\endgroup$
    – Jochen Wengenroth
    Commented Mar 1 at 11:13
  • $\begingroup$ @JochenWengenroth By isomorphism, you mean a surjective linear isometry between Frechet spaces? $\endgroup$
    – Isaac
    Commented Mar 1 at 11:22
  • $\begingroup$ @JochenWengenroth But even with an isometry, I don't clearly see how the "tame structure", as described in tame Frechet spaces and Nash-Moser therem, can be preserved. As you can see, seminorms are involved in the definition. $\endgroup$
    – Isaac
    Commented Mar 1 at 11:24
  • $\begingroup$ You are right -- but you did not specify a grading on$\mathscr S(\mathbb R^N)$. Dietmar Vogt worked a lot about graded Fréchet spaces. $\endgroup$
    – Jochen Wengenroth
    Commented Mar 1 at 13:06
  • $\begingroup$ @JochenWengenroth I already did in the post : the Schwartz space is a graded Frechet space $\endgroup$
    – Isaac
    Commented Mar 1 at 13:46

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