The real H(curl) graph space on a polyhedron

Assume Countable Choice for weak-derivative uniqueness and the real $L^2$ product below, and let $\mathrm{\Omega }\subset {\mathbb{R}}^3$ be open. Here $L^2(\mathrm{\Omega };{\mathbb{R}}^3)$ is the set of triples of real scalar $L^2$ classes from The space $L^p(\mu )$ as the quotient by null functions. Define $(u,v{)}_{L^2(\mathrm{\Omega })}:={\sum }_{j=1}^3{\int }_{\mathrm{\Omega }}{u}_j{v}_{j}dx,\mathrm{\parallel }u{\mathrm{\parallel }}_{L^2(\mathrm{\Omega })}^2:={\sum }_{j=1}^3\mathrm{\parallel }{u}_j{\mathrm{\parallel }}_{L^2(\mathrm{\Omega })}^2\mathrm{.}$ Each scalar pairing is well-defined and induces its quotient norm by $L^2$ with the integral pairing is a Hilbert space; the finite sum is therefore a real inner product on the triple space with the displayed norm. A test field $\phi \in C_c^{\mathrm{\infty }}(\mathrm{\Omega };{\mathbb{R}}^3)$ means a triple of real-valued members of the test-function space $\mathcal{D}(\mathrm{\Omega })$ of Test function space d of an open set. Writing $\phi =({\phi }_x,{\phi }_y,{\phi }_z)$, its classical curl is the coordinate expression of Divergence and curl of a $C^1$ vector field, $\mathrm{curl}\phi =({\mathrm{\partial }}_y{\phi }_z-{\mathrm{\partial }}_z{\phi }_y,{\mathrm{\partial }}_z{\phi }_x-{\mathrm{\partial }}_x{\phi }_z,{\mathrm{\partial }}_x{\phi }_y-{\mathrm{\partial }}_y{\phi }_x)\mathrm{.}$ For $u\in L^2(\mathrm{\Omega };{\mathbb{R}}^3)$, its distributional curl is the triple of scalar distributions whose action on a vector test field is the sum of the three component pairings, determined by $⟨\mathrm{curl}u,\phi ⟩={\int }_{\mathrm{\Omega }}u\cdot \mathrm{curl}\phi dx,\phi \in C_c^{\mathrm{\infty }}(\mathrm{\Omega };{\mathbb{R}}^3)\mathrm{.}$ The integral is finite because $u$ and the smooth compactly supported field $\mathrm{curl}\phi$ are componentwise in $L^2$ and scalar Cauchy–Schwarz applies. On tests supported in a fixed compact $K⋐\mathrm{\Omega }$, the same estimate gives $\mathrm{\mid }⟨\mathrm{curl}u,\phi ⟩\mathrm{\mid }\le C_{u,K}{\max }_{\mathrm{\mid }\alpha \mathrm{\mid }\le 1}{\sup }_K\mathrm{\mid }{D}^{\alpha }\phi \mathrm{\mid }$ with a finite $C_{u,K}$. This proves continuity on each fixed-support test space; the LF universal property of Test function lf topology universal property, for the test topology of Test function topology, makes the pairing a vector distribution on $\mathrm{\Omega }$. If this vector distribution is represented by an $L^2(\mathrm{\Omega };{\mathbb{R}}^3)$ class, that class is unique almost everywhere componentwise by the injection of locally integrable functions into scalar distributions in Locally integrable functions embed in distributions, under the stated Countable Choice assumption. Define $H(\mathrm{curl};\mathrm{\Omega })=\{u\in L^2(\mathrm{\Omega };{\mathbb{R}}^3):\mathrm{curl}u\text{ is represented by an element of }{L}^2(\mathrm{\Omega };{\mathbb{R}}^3)\}$ with graph norm $\mathrm{\parallel }u{\mathrm{\parallel }}_{H(\mathrm{curl};\mathrm{\Omega })}^2=\mathrm{\parallel }u{\mathrm{\parallel }}_{L^2(\mathrm{\Omega })}^2+\mathrm{\parallel }\mathrm{curl}u{\mathrm{\parallel }}_{L^2(\mathrm{\Omega })}^2$. No boundary condition is imposed by this definition. Fields are real-valued; equality is equality of almost-everywhere classes.