I m currently working my way trough comparing some topologies and I dont understand why the following specific form of semi-norms (I mean the '+1') on the description of the Schwartz space usual topology

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

I read that this induce the same l.c. topology on 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 do not find any clarification about the advantages (interest) of the first form in textbooks. Any guidance on this would be greatly appreciated!



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