The zero-boundary Sobolev energy space is a separable Hilbert space

If $\mathrm{\Omega }=\mathrm{\varnothing }$, then $V=H=\{0\}$ and the assertions hold. Otherwise choose finite $a<b$ such that $\mathrm{\Omega }\subset \{x:a<x_1<b\}$; boundedness permits this. Then [F4] applies to $V$ and gives $\mathrm{\parallel }v{\mathrm{\parallel }}_H\le C_{\mathrm{\Omega }}\mathrm{\parallel }\mathrm{\nabla }v{\mathrm{\parallel }}_2$.

The rational box-step family from [F2], restricted to $\mathrm{\Omega }$, is countable and dense in $H$: for $h\in H$, set $Eh=h$ on $\mathrm{\Omega }$ and $Eh=0$ elsewhere. This extension is measurable because $\mathrm{\Omega }$ is measurable, and ${\int }_{{\mathbb{R}}^d}\mathrm{\mid }Eh{\mathrm{\mid }}^2={\int }_{\mathrm{\Omega }}\mathrm{\mid }h{\mathrm{\mid }}^2$. Extension of a null-set difference remains null, so $E$ is well-defined on $L^2$ classes and is an isometry. Approximate $Eh$ in $L^2({\mathbb{R}}^d)$ by [F2], and restrict the approximants back to $\mathrm{\Omega }$, which cannot increase the $L^2$ error. Therefore $H$ is separable by [F7]. Finite products of the resulting countable dense set show that $Z:=H^{d+1}$ is separable; its product inner product makes it a real Hilbert space.

Map $J:W^{1,2}(\mathrm{\Omega };\mathbb{R})\to Z$ by $J(u)=(u,D_{1}u,\dots ,D_{d}u)$. The norm from [F3] makes $J$ an isometry onto its image. Its image is closed: if $J(u_n)\to (u,g_1,\dots ,g_d)$ in $Z$, then for every $\phi \in C_c^{\mathrm{\infty }}(\mathrm{\Omega })$ and every $i$, ${\int }_{\mathrm{\Omega }}u{\mathrm{\partial }}_i\phi ={\lim }_n{\int }_{\mathrm{\Omega }}{u}_n{\mathrm{\partial }}_i\phi =-{\lim }_n{\int }_{\mathrm{\Omega }}{D}_i{u}_n\phi =-{\int }_{\mathrm{\Omega }}{g}_i\phi ,$ where each limit follows from Cauchy–Schwarz in $L^2$. Thus $g_i$ is the weak derivative $D_{i}u$, and the limit lies in the image. A closed subspace of the complete Hilbert space $Z$ is complete in its inherited inner product, so $W^{1,2}(\mathrm{\Omega })$ is Hilbert.

By [F3], $V$ is a closed subspace of $W^{1,2}(\mathrm{\Omega })$, hence Hilbert for the inherited Sobolev inner product. From step 1.1, $\mathrm{\parallel }\mathrm{\nabla }v{\mathrm{\parallel }}_2^2\le \mathrm{\parallel }v{\mathrm{\parallel }}_{W^{1,2}}^2=\mathrm{\parallel }v{\mathrm{\parallel }}_2^2+\mathrm{\parallel }\mathrm{\nabla }v{\mathrm{\parallel }}_2^2\le (1+C_{\mathrm{\Omega }}^2)\mathrm{\parallel }\mathrm{\nabla }v{\mathrm{\parallel }}_2^2\mathrm{.}$ Thus the gradient norm is equivalent to the inherited Hilbert norm and is also complete; the norm chosen for $V$ in the Statement is the inherited $W^{1,2}$ norm.

Since $W^{1,2}(\mathrm{\Omega })$ is a subspace of the separable metric space $Z$, it is separable. For any point $x$ in a relatively open set, choose $ϵ>0$ such that $B(x,ϵ)\cap W^{1,2}$ lies in that set. Choose a center $z$ from a countable dense subset of $Z$ with $\mathrm{\parallel }x-z\mathrm{\parallel }<ϵ\mathrm{/}4$, then a positive rational $r$ with $\mathrm{\parallel }x-z\mathrm{\parallel }<r<ϵ-\mathrm{\parallel }x-z\mathrm{\parallel }$. The relative balls $B(z,r)\cap W^{1,2}$ form a countable base. By [A1], choose one point from every nonempty base member; these points are dense. Apply the same argument to the subspace $V$. The equivalent gradient norm from step 3.1 induces the same topology, so $V$ is separable in its stated norm.

By [F5], $C_c^{\mathrm{\infty }}(\mathrm{\Omega };\mathbb{R})$ is dense in $H$. It is a subset of $V$ by [F3], so $V$ is dense in $H$. Step 1.1 gives continuity of the inclusion, and injectivity holds because both spaces consist of almost- everywhere classes of the same functions. The defining hypotheses in [F6] now hold, giving the canonical dense inclusion $H\hookrightarrow V^{\ast }$ and the claimed evolution triple.