Ultraweak Poisson skeleton spaces are Hilbert trace quotients

H1 is complete: completeness of real $L^2$ [F6] gives limits in $L^2$ of a Cauchy sequence and each weak derivative satisfy distributional derivative identities by testing against compactly supported smooth functions and passing both integrals to their limits. H1_0 is closed by its closure definition. The scalar trace-quotient map is continuous because $\mathrm{\parallel }[u{\mathrm{\mid }}_K]{\mathrm{\parallel }}_{H^1(K)\mathrm{/}{H}_0^1(K)}\le \mathrm{\parallel }u{\mathrm{\mid }}_K{\mathrm{\parallel }}_{H^1(K)}$ on each cell; the finite product preserves continuity. The normal map is continuous by [F2], again with a finite product. Thus both kernels are closed.

Let $\tau \in H(\mathrm{div};\mathrm{\Omega })$ and let an interior facet $F=K_{+}\cap K_{-}$ be fixed. For $\psi \in C_c^{\mathrm{\infty }}(\mathrm{relint}F)$, choose a smooth function $\varphi \in C_c^{\mathrm{\infty }}(\mathrm{\Omega })$ supported in a small tubular neighborhood of $\mathrm{supp}\psi$ meeting only $K_{+}$ and $K_{-}$ and having trace $\psi$ on $F$. The global distributional-divergence identity [F8] gives ${\int }_{\mathrm{\Omega }}(\tau \cdot \mathrm{\nabla }\varphi +(\mathrm{div}\tau )\varphi )=0$. Applying the cell Green identity [F2] to the two restrictions, with all other cell traces zero, yields $⟨{\gamma }_{n_{+},K_{+}}(\tau {\mathrm{\mid }}_{K_{+}}),[\varphi {\mathrm{\mid }}_{K_{+}}]⟩+⟨{\gamma }_{n_{-},K_{-}}(\tau {\mathrm{\mid }}_{K_{-}}),[\varphi {\mathrm{\mid }}_{K_{-}}]⟩=0.$ The two cell trace classes in this equation restrict to the same $\psi$ on $F$ and vanish on the other facets. Since this holds for every such $\psi$, the two outward weak normal traces cancel as distributions on the relative interior of each interior facet. This is the conservative facet relation for a global H(div) lift; it uses the distributional divergence identity and weak Green formula, not pointwise normal values.

For a closed subspace $M$ of a Hilbert space, a distance-minimizing sequence is Cauchy by the parallelogram identity; its limit is nearest, and variation along $M$ makes the residual orthogonal. Thus $H=M\oplus M^{\perp }$ and $H\mathrm{/}M$ is isometric to $M^{\perp }$. Applying this to both kernels in step 1.1 proves the quotient ranges are Hilbert with their infimum norms.

Global lifts restrict to broken lifts; conversely the admissible broken extensions in the Statement are exactly those gluing to a global lift, so the two infima have the same set. Cell scalar traces are bounded by [F3] and those of a global H1 lift match across interior facets by [F4]; normal traces of a global H(div) lift satisfy the cancellation proved in step 1.2. Thus range membership is precisely the stated compatibility, with the global lift infimum norms.