Lemma 9

A linear mapping $f$ of $\mathscr{S}(\mathbf{R}^n)$ into $\mathbf{C}$ is a tempered distribution if and only if for every family $(M_{\alpha ,k})_{(\alpha ,k)\in\mathbf{N}^n\times\mathbf{N}}$ in $\mathbf{R}_+$, the linear form $f$ is bounded on the set of functions $\varphi \in \mathscr{S}(\mathbf{R}^n)$ such that $\widetilde{q}_{\alpha ,k}(\varphi )\leqslant M_{\alpha ,k}$ for every $(\alpha , k)\in \mathbf{N}^n\times \mathbf{N}$.

In fact, the space $\mathscr{S}(\mathbf{R}^n)$ is bornological (EVT, III, p. 12, prop. 2), and every bounded subset of $\mathscr{S}(\mathbf{R}^n)$ is contained in one of the bounded sets described in the statement.