I want to make a statement about the following sequence of functions whether it converges in $\mathcal{D}'(\mathbb{R})$ (space of distributions): \begin{align} \phi_n(x) = e^{-x^2/n^2}. \end{align} I want to do it as analytical as possible.
The definition is clear but I do not see how to applicate it to our problem:
$T_n \rightarrow T$ in $\mathcal{D}'(\mathbb{R})$ if $T_n(f) \rightarrow T(f)$ for all $f \in \mathcal{D}(\mathbb{R})$.
Is there anyone who can help me :)
For any $f \in \mathcal{D}(\mathbb{R})$ $$ \langle\phi_n,f\rangle=\int_{\Bbb R}e^{-x^2/n^2}\,f(x)\,dx. $$ Use the dominated convergence theorem to show that the integral converges as $n\to\infty$.