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I have big problems proving that the Schwartz Space $\mathscr{S}(\mathbb{R}^n)$ together with the topology induced by the family $$ \|\varphi\|_{p}:=\sup_{x\in \mathbb{R^n}}\sup_{|\alpha|\leq p}|(1+|x|^2)^pD^\alpha \varphi(x)| $$ is a nuclear F-space. I am trying to find a proof on an elementary level, i.e. using the characterisation via Hilbert-Schmidt/nuclear operators. I managed to prove, that $\mathscr{S}$ is nuclear with the family $$ \|\varphi\|_{p}:=\sup_{x\in \Omega_p}\max_{|\alpha|\leq p}|(1+|x|^{2})^{p}D^{\alpha}\varphi(x)| $$ for some bounded upwardly directed $\Omega_p$, but unfortunately, this topology is too weak and the generalization for unbounded $\Omega_p$ is not possible (I slightly generalized the Sobolev Emedding theorem here, but unfortunately, I need bounded regions to do this).

I also tried doing stuff with Hermite Polynomials, but that didn't lead to much either because I couldn't properly estimate them. The only proofs I found in books were using other characterizations which I don't want to introduce.

Does anyone know how this works? I would greatly appreciate any help.

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The map $\mathbb R^n\to S^n$ by $x\to ({x\over \sqrt{1+|x|^2}},\,{1\over \sqrt{1+|x|^2}})$ sets up L. Schwartz' characterization of the Schwartz functions as smooth functions on $S^n$ vanishing to infinite order at the point at infinity.

If we know that smooth functions on $S^n$ or other compact Riemannian manifolds are nuclear Frechet, then the same (can be proven as an exercise) to hold for closed subspaces.

For $S^n$, the explicit decomposition of $L^2$ in terms of spherical harmonics (and various of their standard properties) can give a more concrete/explicit proof of the nuclear-Frechet-ness.

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