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Good morning, I'm studying quantum mechanics as a mathematician. I read that the Schwartz class $$\begin{equation*} \mathcal{S}(\mathbb{R}) := \{ \varphi \in \mathcal{C}^\infty (\mathbb{R}, \mathbb{C}) : \forall p,q \in \mathbb{N} \quad \sup_{x \in \mathbb{R}} |x^p \, \partial^q \varphi(x)| < +\infty \} \end{equation*}$$ is a nuclear space. I can't find a good textbook where to read a proof of this statement. Does anyone know one that has, with a student friendly proof? Thanks in advance.

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  • $\begingroup$ This is treated in Chapter 3 of Quantum Mechanics in Rigged Hilbert Space Language by Rafeal de la Madrid $\endgroup$
    – whpowell96
    Commented Apr 8 at 17:49
  • $\begingroup$ Greatly appreciated, thanks $\endgroup$
    – Marco Lugarà
    Commented Apr 29 at 9:21

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