Let $Af=f(0)$ be a linear form restricted to the space of $C^{\infty}$-funtions with compact support over $\mathbb{R}$ (noted $C_c^{\infty}(\mathbb{R})$). Is $A$ continuous w.r.t. the topology of $L^2(\mathbb{R})$? In other words, can we find a sequence $(f_n)$ in $C_c^{\infty}(\mathbb{R})$ such that $\|f_n-f\|_2\to 0$ with $f_n(0)\not\to f(0)$ where $f\in C_c^{\infty}(\mathbb{R})$?
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Math
Yes, there are such sequences. For instance, if $$g(x)=\begin{cases}\exp\frac{1}{x^2-1}&\text{if }x\in (-1,1)\\ 0&\text{if }x\notin (-1,1)\end{cases}$$
Then $f_n(x)=g(nx)$ converges in $L^2$ to $0$, and specifically $$\left\lVert f_n-0\right\rVert_2=n^{-1/2}\sqrt{\int_{\Bbb R} (g(x))^2\,dx}.$$ Yet, $f_n(0)=\frac1e$.