Lemma: Finite linear probes leave an infinite-dimensional blind sector

Let $W\subset {\mathbb{R}}^d$ be nonempty and open, with $d\ge 1$. For any $m$ linear functionals ${\ell }_1,\dots ,{\ell }_m$ on ${C_c}^{\infty }(W)$, there is a nonzero $k\in {C_c}^{\infty }(W)$ with ${\ell }_i(k)=0$ for every $i$.

Proof. Choose $m+1$ nonzero smooth functions with pairwise disjoint compact supports in $W$. Their span has dimension $m+1$. The evaluation map from that span to ${\mathbb{C}}^m$ has a nonzero kernel.