The restriction distribution is zero implies that the distribution is zero (Theorem 2.2.1 Hormander)

Question

Hello. Why $u(\phi_j)=0$? I can't see this part.

Answer 1

The condition that $u(\phi_j)=0$ is built into the hypotheses. Indeed, the assumption is that every point has a neighborhood in which the restriction of $u$ is $0$. The key step in the proof is the use of "...we can choose a finite number of SUCH open sets..." with the "such" referring to open sets on which the restriction of $u$ vanishes. So then the point is that $\phi_j$ is supported in an open set in which $u$ vanishes, and so the identity $u(\phi_j)=0$ then follows by definition.