We call a distribution on $\mathcal{D}$ any linear map $T : \mathcal{D} \rightarrow \mathbb{R}$ which satisfies $$\forall \varphi \in \mathcal{D}, \forall (\varphi_n)_{n \in \mathbb{N}} \in \mathcal{D}^{\mathbb{N}} \quad \varphi_n \xrightarrow{\mathcal{D}} \varphi \Longrightarrow T(\varphi_n) \rightarrow T(\varphi)$$ Show that if $f \in \mathcal{F}_{sr}$ then the map $T_f$ defined by $$\forall \varphi \in \mathcal{D} \quad T_f(\varphi) = \int_{-\infty}^{+\infty} f(x) \varphi(x) \mathrm{d}x$$ defines a distribution on $\mathcal{D}$.
We call a distribution on $\mathcal{D}$ any linear map $T : \mathcal{D} \rightarrow \mathbb{R}$ which satisfies
$$\forall \varphi \in \mathcal{D}, \forall (\varphi_n)_{n \in \mathbb{N}} \in \mathcal{D}^{\mathbb{N}} \quad \varphi_n \xrightarrow{\mathcal{D}} \varphi \Longrightarrow T(\varphi_n) \rightarrow T(\varphi)$$
Show that if $f \in \mathcal{F}_{sr}$ then the map $T_f$ defined by
$$\forall \varphi \in \mathcal{D} \quad T_f(\varphi) = \int_{-\infty}^{+\infty} f(x) \varphi(x) \mathrm{d}x$$
defines a distribution on $\mathcal{D}$.