Kernel of multiplication by a coordinate
= Kernel of multiplication by a coordinate
In one dimension, $xw=0$ if and only if $w=c\delta_0$. Choose a <cutoff function> $\eta=1$ near zero and write every <test function> as $\varphi=\varphi(0)\eta+x\psi$. The pairing with $x\psi$ vanishes, leaving $\langle w,\varphi\rangle=\varphi(0)\langle w,\eta\rangle$.