= Schwartz seminorm
{c}
{title2=$p_{\alpha\beta}(\varphi)=\sup_x|x^\alpha\partial^\beta\varphi(x)|$}
These <seminorms> measure all polynomially weighted <derivatives> and generate the Fréchet topology of the <Schwartz space>. Finitely many such bounds control each weighted <derivative>'s integrable norm by adding a spatial decay factor stronger than $\langle x\rangle^{-n}$. <Continuity> of the <Fourier transform> follows by applying those bounds after differentiation and <integration by parts>. Equivalent families use powers of $\langle x\rangle$ instead of individual monomials.
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