Countable smooth solenoidal basis for bounded closure spaces

The zero-extension map embeds $H$ isometrically into $L^2$ of a periodic cube containing $\bar{\mathrm{\Omega }}$ in its interior. By [F2], it embeds $V$ into the finite product consisting of the zero extension and its three first derivatives in $L^2$ of that cube. Each ambient space is separable by [F4]. Every subspace of a separable metric space is separable: take its countable base of intersections with ambient rational balls and, using [F5], choose one point from each nonempty intersection. Thus $H$ and $V$ are separable.

The $L^2$ and $H^1$ graph inner products make the ambient spaces Hilbert: completeness of $L^2$ and the Sobolev completeness in [F3] give convergence of Cauchy sequences in the function and all derivative coordinates, and the weak-derivative identity passes to the limit against each test. The spaces $H,V$ are closed by their definitions as closures, so they inherit Hilbert structure. The $V$ norm dominates the $H$ norm; since $D_{\sigma }$ is dense in both closures, the inclusion $V\hookrightarrow H$ is continuous with dense image.

Separability of $V$ and density of $D_{\sigma }$ in $V$ allow Countable Choice to select $d_j\in D_{\sigma }$ approximating a countable dense subset of $V$ to every rational precision; the selected countable collection is dense in $V$ and, because $D_{\sigma }$ is also dense in $H$, its linear span is dense in $H$. The space $D_{\sigma }$ is infinite dimensional: choose countably many disjoint balls in one ball contained in $\mathrm{\Omega }$ and in each choose a nonzero smooth compactly supported curl field; their disjoint supports give linear independence. Apply Gram–Schmidt in $H$ to the selected collection, discarding zero remainders. Each resulting $e_j$ is a finite linear combination of elements of $D_{\sigma }$, so is still in $D_{\sigma }$; its span contains every selected $d_j$ after the corresponding finite stage and is therefore dense in $V$ and $H$.