Lemma 10

Let $\mathfrak{F}$ be a filter on $\mathscr{S}'(\mathbf{R}^n)$ having a countable base or containing a simply bounded set. Then $\mathfrak{F}$ converges to a tempered distribution if and only if $\langle f, \varphi \rangle$ converges along $\mathfrak{F}$ for every Schwartz function $\varphi$.

In particular, 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 $\varphi \in \mathscr{S}(\mathbf{R}^n)$.

Since $\mathscr{S}'(\mathbf{R}^n)$ is a Fréchet space, hence barrelled, (EVT, III, p. 25, cor. of prop. 2) the lemma results from the Banach-Steinhaus theorem (EVT, III, p. 26, cor. 3 of th. 1).

The canonical injection $j$ of $\mathscr{D}(\mathbf{R}^n)$ into $\mathscr{S}(\mathbf{R}^n)$ is continuous, and its image is dense (lemma 12 of IV, p. 212); hence the transpose of $j$, which is the restriction mapping of tempered distributions to the subspace $\mathscr{D}(\mathbf{R}^n)$, is an injective continuous linear mapping of $\mathscr{S}'(\mathbf{R}^n)$ into $\mathscr{D}'(\mathbf{R}^n)$. We shall identify $\mathscr{S}'(\mathbf{R}^n)$ with a subspace of $\mathscr{D}'(\mathbf{R}^n)$ by means of this mapping.

Let $\alpha \in \mathbf{N}^n$. The linear mapping $\varphi \mapsto \partial^{\alpha}\varphi$ of $\mathscr{S}(\mathbf{R}^n)$ into $\mathscr{S}(\mathbf{R}^n)$ is continuous. Its transpose is therefore a continuous linear mapping of $\mathscr{S}'(\mathbf{R}^n)$ into $\mathscr{S}'(\mathbf{R}^n)$ (EVT, IV, p. 6, cor., b)). The continuous linear mapping $(-1)^{|\alpha|t}\partial^{\alpha}$ of $\mathscr{S}'(\mathbf{R}^n)$ into $\mathscr{S}'(\mathbf{R}^n)$ is denoted by $\partial^{\alpha}$. This definition is compatible with Definition 2 of IV, p. 208 for distributions.

Let $h\in \mathscr{C}^{\infty}(\mathbf{R}^n)$ be a function such that $\partial^{\alpha}h$ has polynomial growth for every $\alpha \in \mathbf{N}^n$. The transpose of the continuous linear mapping $\varphi \mapsto h\varphi$ (prop. 14 of IV, p. 214) is a continuous linear mapping on $\mathscr{S}'(\mathbf{R}^n)$, denoted by $f\mapsto hf$.