The function space of rapidly decreasing functions $\mathcal{S}$ has the important property that the Fourier transform is an endomorphism on this space. This property enables one, by duality, to define the Fourier transform for elements in the dual space of $\mathcal{S}$, that is, for tempered distributions.

Definition The space of rapidly decreasing functions on $\mathbb{R}^{n}$ is the function space

where $C^{\infty}(\mathbb{R}^{n})$ is the set of smooth functions from $\mathbb{R}^{n}$ to $\mathbb{C}$, and

Here, $||\cdot||_{\infty}$ is the supremum norm, and we use multi-index notation. When the dimension $n$ is clear, it is convenient to write $\mathcal{S}=\mathcal{S}(\mathbb{R}^{n})$. The space $\mathcal{S}$ is also called the Schwartz space, after Laurent Schwartz (1915-2002) [2].