Lemma: Finite linear tests leave smooth kernel ambiguity

Let $W$ be a nonempty open set, and let ${\ell }_1,\dots ,{\ell }_m$ be linear functionals on ${C_c}^{\infty }(W)$. There is a nonzero smooth compact function $k$ with ${\ell }_i(k)=0$ for all $i$. Thus finitely many linear kernel evaluations do not determine an unrestricted compact smooth kernel.

Proof. Choose $m+1$ nonzero smooth functions with pairwise disjoint compact supports in $W$. They are linearly independent. The map from their span to ${\mathbb{C}}^m$ given by the ${\ell }_i$ has a nonzero kernel. Any nonzero element of that kernel is the required $k$.