If $u$ is a tempered distribution and $((1+\xi^2)u,\phi)=0$ for all $\phi$ Schwartz function. Then $u$ is the zero tempered distribution?

Actualization. I think that I proves this. My attempt: $\left\{\phi\in \mathcal{S}\right\}=\left\{\psi/(1+\xi^2):\psi\in\mathcal{S}\right\}$ coincide. Now, $((1+\xi^2)u,\phi)=(u,(1+\xi^2)\phi)=(u,\psi)=0$ for all $\psi$ Schwartz function. Now, the fourier transform is an isomorphism on the tempered distribution space, then $u=0$. This is correct? Thanks.



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