Consider the following family of semi-norms on the Schwartz space

$$\|f\|_{m,n}=\sup_{x\in \mathbb{R}}|(1+|x|)^m f^{(n)}(x)| \;\;\; m,n\in \{0,1,2,...\} $$

It is well known in the litterature that the family above induce the same locally convex topology on the Schwartz space as the family

$$\|f\|_{m,n}=\sup_{x\in \mathbb{R}}|x^m f^{(n)}(x)| \;\;\; m,n\in \{0,1,2,...\} $$

I can not figure out what is the advantages/interests of using the first family of semi-norms instead of the second. Any guidance on this would be greatly appreciated!

  • $\begingroup$ The sequence of weights $(1+|x|)^m$ is increasing. $\endgroup$
    – Jochen
    Aug 30, 2022 at 5:19
  • $\begingroup$ thank you so much. $\endgroup$
    – user536450
    Aug 30, 2022 at 8:49


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