Lemma 4

Let $f$ be a linear mapping of $\mathscr{D}(U)$ into $\mathbf{C}$. Then $f$ is a distribution if and only if, for every compact subset K of U and every family $(M_{\alpha})_{\alpha\in\mathbf{N}^n}$ in $\mathbf{R}_+$, the linear form $f$ is bounded on the set of functions $\varphi \in \mathscr{C}_K^{\infty}(U)$ such that, for every $\alpha \in \mathbf{N}^n$, one has

sup$_{x\in K}|\partial^{\alpha}\varphi (x)|\leqslant M_{\alpha}$.

In fact, the space $\mathscr{D}(U)$ is bornological and every bounded subset of $\mathscr{D}(U)$ is contained in one of the bounded sets described in the statement.