Suppose that $f_{n}(x)$ are a sequence of $L^{2}$ functions which converge to a function $f(x)$ in the $L^{2}$ sense. Show that it also converges weakly in the sense of distributions, ie for any test function $\varphi$ we have that
$\int\limits_{-\infty}^{\infty} f_{n}(x) \varphi(x) dx \rightarrow \int\limits_{\infty}^{\infty} f(x)\varphi(x)dx$.
I recognize that this would be true if the convergence was uniform, but is it necessarily true with only $L^{2}$ convergence?
Attempt at a solution -
$|\int\limits_{-\infty}^{\infty} f_{n}(x) \phi(x)dx - \int\limits_{-\infty}^{\infty} f(x) \phi(x)dx| \leq \int\limits_{-\infty}^{\infty} | \phi(x) | | f_{n}(x) - f(x) | dx $