Lemma: Continuity and independence of normal jets

For each $\alpha$, (eq:Ejet) defines a continuous map

For any finite $m$, the map

is injective.

If $b_{\alpha }$ is supported in a set $L\subset S_0$, its insertion is supported in $L\times \{0\}$.

Proof. The map $\phi \mapsto {{\partial }_n}^{\alpha }\phi (s,0)$ is continuous from compactly supported tests in $U$ to compactly supported tests in $S_0$. Transposition defines (eq:Ejet) continuously and gives its support.

For independence choose $\chi (n)$ compactly supported and equal to one near zero, and use the tests

For every $|\alpha |\le m$,

These tests recover each $b_{\gamma }$ separately.