Definition 12.4.3 (Schwartz Class).label Denote the Schwartz Class by
\begin{align*}\cali{S}(\R^{n})=\curl{u\in C^\infty(\R^n):\SUP{x\in\R^n}\abs{x^\al\pd^\be u}<\infty}\end{align*}
Since $C^{\infty}_{c}(\R^{n})\suf\cali{S}(\R^{n})$, we have $\cl{\cali{S}(\R^n)}=L^{p}(\R^{n})$ for all $p\geq 1$ under the $L^{p}(\R^{n})$ topology. In particular $\cali{S}(\R^{n})$ are uniformly continuous.
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