Definition 6

The space of tempered distributions on $\mathbf{R}^n$ is, by definition, the dual space of $\mathscr{S}(\mathbf{R}^n)$ endowed with the topology of bounded convergence. It is denoted by $\mathscr{S}'(\mathbf{R}^n)$.

Since $\mathscr{S}(\mathbf{R}^n)$ is bornological, the space $\mathscr{S}'(\mathbf{R}^n)$ is complete and bornological (EVT, III, p. 24, cor. 1 and 2). Since $\mathscr{S}(\mathbf{R}^n)$ is a Montel space, the same is true of $\mathscr{S}'(\mathbf{R}^n)$ (EVT, IV, p. 19, prop. 9).

Let $\alpha \in \mathbf{N}^n$. The transpose of the endomorphism $\varphi \mapsto X^{\alpha}\varphi$ of $\mathscr{S}(\mathbf{R}^n)$ is again denoted by $f\mapsto X^{\alpha}f$, and $f\mapsto \partial^{\alpha}f$ denotes the endomorphism of $\mathscr{S}'(\mathbf{R}^n)$ defined by

for $f\in \mathscr{S}'(\mathbf{R}^n)$ and $\varphi \in \mathscr{S}(\mathbf{R}^n)$.

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

A sequence $(f_m)_{m\in\mathbf{N}}$ of tempered distributions converges to a tempered distribution $f$ if, and only if, one has $\langle f_m, \varphi \rangle \rightarrow \langle f, \varphi \rangle$ for every $\varphi \in \mathscr{S}(\mathbf{R}^n)$.