= Fourier transform isomorphism of the Schwartz space
{c}
{title2=$\mathcal F:\mathcal S\longrightarrow\mathcal S$}
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 $\mathcal Rf(x)=f(-x)$. Transposition gives a continuous isomorphism on <tempered distributions> as well.
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