The Fourier transform is a linear isomorphism $\mathcal{S}(\mathbb{R}^n)\to\mathcal{S}(\mathbb{R}^n)$

The Fourier transform $(\mathcal{F}f)(\xi)=\int_{\mathbb{R}^n} e^{-2\pi i\langle x,\xi\rangle}f(x)\,dx$ is a linear isomorphism of $\mathcal{S}(\mathbb{R}^n)$ onto itself: it maps $\mathcal{S}(\mathbb{R}^n)$ into $\mathcal{S}(\mathbb{R}^n)$, it is injective on $\mathcal{S}(\mathbb{R}^n)$, every element of $\mathcal{S}(\mathbb{R}^n)$ is the Fourier transform of some element of $\mathcal{S}(\mathbb{R}^n)$, and $\mathcal{F}(f+cg)=\mathcal{F}f+c\,\mathcal{F}g$ for all $f,g\in\mathcal{S}(\mathbb{R}^n)$, $c\in\mathbb{C}$.