The restricted Matérn native space is the Sobolev quotient space

The set ${\mathcal{N}}_{\mathrm{\Omega }}$ is a linear subspace. If $U_n\in {\mathcal{N}}_{\mathrm{\Omega }}$ converges to $U$ in native norm, then for each $x\in \mathrm{\Omega }$, [F2] gives $\mathrm{\mid }{U}_n(x)-U(x)\mathrm{\mid }\le C_x\mathrm{\parallel }{U}_n-U{\mathrm{\parallel }}_k\to 0$; hence $U(x)=0$ and $U\in {\mathcal{N}}_{\mathrm{\Omega }}$. Thus ${\mathcal{N}}_{\mathrm{\Omega }}$ is closed.

By [F3], each equivalence class modulo ${\mathcal{N}}_{\mathrm{\Omega }}$ has a unique representative in ${\mathcal{N}}_{\mathrm{\Omega }}^{\perp }$, namely its orthogonal projection onto that complement; its norm is the infimum of the norms of all extensions because adding an element of ${\mathcal{N}}_{\mathrm{\Omega }}$ changes only the nullspace component. Therefore the restriction quotient is a Hilbert space isometric to ${\mathcal{N}}_{\mathrm{\Omega }}^{\perp }$.

For $y\in \mathrm{\Omega }$, the section $k_y=k(\cdot -y)$ lies in ${\mathcal{N}}_{\mathrm{\Omega }}^{\perp }$: for every $N\in {\mathcal{N}}_{\mathrm{\Omega }}$, $⟨N,k_y{⟩}_k=N(y)=0$ by reproduction [F2]. Its induced quotient element therefore represents the restricted kernel section, and it reproduces evaluation of each restricted function at $y$.

If $U\in {\mathcal{N}}_{\mathrm{\Omega }}^{\perp }$ is orthogonal to every $k_y$ with $y\in \mathrm{\Omega }$, then [F2] gives $U(y)=0$ for all $y\in \mathrm{\Omega }$, so $U\in {\mathcal{N}}_{\mathrm{\Omega }}$. Thus $U\in {\mathcal{N}}_{\mathrm{\Omega }}\cap {\mathcal{N}}_{\mathrm{\Omega }}^{\perp }=\{0\}$. The restricted kernel sections span densely in ${\mathcal{N}}_{\mathrm{\Omega }}^{\perp }$, which proves that the quotient is exactly the RKHS of the restricted kernel.

Every extension $U$ of $u$ satisfies $c_0\mathrm{\parallel }U{\mathrm{\parallel }}_{H^s({\mathbb{R}}^d)}\le \mathrm{\parallel }U{\mathrm{\parallel }}_k\le C_0\mathrm{\parallel }U{\mathrm{\parallel }}_{H^s({\mathbb{R}}^d)}$ by [F1]. Taking the infimum over the same set of extensions on both sides gives $c_0\mathrm{\parallel }u{\mathrm{\parallel }}_{H^s(\mathrm{\Omega })}\le \mathrm{\parallel }u{\mathrm{\parallel }}_{{\mathcal{H}}_k(\mathrm{\Omega })}\le C_0\mathrm{\parallel }u{\mathrm{\parallel }}_{H^s(\mathrm{\Omega })}$.