Lemma: Injectivity of the generating test maps

The maps $f\mapsto B_f$ and $g\mapsto W_g$ are injective.

Proof. Injectivity of $f\mapsto B_f$ is IX-A's result IXA (cor:notbulk).

Suppose $W_g=0$. Then $\int g{v}^2=0$ for every smooth scalar function $v$, by taking $h_{00}=v$. Polarization gives $\int guv=0$ for every smooth $u,v$. Taking $v=1$ proves that $g$ is the zero distribution, hence the zero test.