Properties of $L^2(-1,1)$ functions I want to show that there is no function $v \in L^2(-1,1)$ with $\int_{-1}^{1} v(x)\phi(x) dx = 2\phi(0)$ for all $\phi \in C^\infty_0(-1, 1)$ ($\phi$ is $0$ everywhere but $[-1,1] $).
I know about the delta distribution or the dirac measure, but I'd like to solve this without using either of these, if possible. I'm pretty helpless because the condition that $\phi \in C^\infty_0(-1, 1)$ is pretty strong, so I can't construct any counterexamples.

By Cauchy-Schwarz,
$$
2\phi(0)\le\|\nu\|_2\cdot\|\phi\|_2.
$$
Can you find $\phi\in C^\infty_0(-1,1)$ with $\phi(0)=1$ but $\|\phi\|_2$ very small — say smaller than $2/\|\nu\|_2$?
