 Fourier transform isomorphism of the Schwartz space

The Fourier transform with negative exponential and inverse factor $(2\pi {)}^{-n}$ is a continuous isomorphism of the Schwartz space. Weighted frequency derivatives are transforms of derivatives of weighted input functions, so their suprema are controlled by finitely many Schwartz seminorms. Fourier inversion gives ${\mathcal{F}}^2=(2\pi {)}^n\mathcal{R}$, where $\mathcal{R}f(x)=f(-x)$. Transposition gives a continuous isomorphism on tempered distributions as well.