Lemma: Compact dual family

There is a nonnegative $\rho \in {C_c}^{\infty }((t_0,T)\times {\mathbb{R}}^3)$ for which

is positive definite. Defining

gives $J_a\in \mathcal{E}$ and $F_{J_a}(v_c)={\delta }_{ac}$.

Proof. The two curvature fields are linearly independent on the open slab. Otherwise a nonzero combination $v$ would satisfy $\Delta v_{ij}=0$ there. Each slice is Schwartz, so $v=0$ on that slab. Wave uniqueness would make it zero everywhere, contrary to independence of $e,r$.

Component evaluations at points in the slab therefore separate the two-dimensional coefficient space. Select a finite separating set and small nonnegative normalized bumps near its points. The sum of the bumps gives a Gram matrix tending to a positive definite sum of squares of separating evaluations, with any additional tensor-component squares nonnegative. A sufficiently small fixed choice is positive definite. Equation (eq:curvaturepairing) then gives the stated duality.