I'm currently reading some features about the Schwartz space $\mathcal{S}(\mathbb{R})$ and I found on a paper of J.E.Roberts that the Schwartz topology on $\mathcal{S}(\mathbb{R})$ i.e. the one generated by the family of inner products $$ ||f||_{m,n}:=\sup_{\mathbb{R}}\left\lvert x^mf^{(n)}(x)\right\rvert ;\;\;m,n\in \mathbb{N} $$ can be seen as generated by the following family of inner products $$ \langle f,g\rangle_n=\int (1+|x|)^n\sum_{|k|\leq n}f^{k}(x)g^{k}(x)dx. $$ Any explanation why this holds (or some references where can i find one) would be greatly appreciated.



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