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!