M$, must form a basis for ${\bf X}_h(\Omega)$ if they are linearly independent. Indeed, by substituting $\chi_h=\omega_{j,h}$ into , we obtain $$\begin{aligned}
\|\nabla \omega_{j,h}\|_{L^2}\leq
C\|{\bf w}_j-\widetilde\Pi_h^{\mathbb N} {\bf w}_j\|_{L^2}
\leq Ch^{1/2+\delta_*}\|{\bf w}_j\|_{H^{1/2+\delta_*}} .
$$ Using the inverse inequality, we see that for $\delta<3\delta_*/(1-\delta_*)$ there holds $$\begin{aligned}
\|\nabla \omega_{j,h}\|_{L^{3+\delta}}\leq
Ch^{-1/2-\delta/(3+\delta)}
\|\nabla \omega_{j,h}\|_{L^2}
\leq Ch^{\delta_*-\delta/(3+\delta)} \rightarrow 0
\quad\mbox{as}\,\,\, h\rightarrow 0 .\end{aligned}$$ Using Lemma \[SmoothPr\] and Lemma \[RegHarV\], we have $$\begin{aligned}
\|{\bf w}_{j,h}-{\bf w}_j\|_{L^{3+\delta}}
&\leq
\|\nabla \omega_{j,h}\|_{L^{3+\delta}}
+\|\widetilde\Pi_h^{\mathbb N} {\bf w}_j-{\bf w}_j\|_{L^{3+\delta}}\\
&\leq Ch^{\delta_*-\delta/(3+\delta)}\|{\bf w}_j\|_{H^{1/2+\delta_*}} 
 \rightarrow 0\quad \mbox{as}\,\,\,h\rightarrow 0,
&& j=1,\cdots,\frak M. \end{aligned}$$ Since ${\bf w}_j$, $j=1,\cdots,\frak M$, are linearly independent and ${\bf w}_{j,h}$ converges to ${\bf w}_{j}$, there exists a positive constant $h_0$ such that ${\bf w}_{j,h}$, $j=1,\cdots,\frak M$, are also linearly independent when $h<h_0$.

A Gram–Schmidt orthogonalization process gives an orthogonal basis which still converges to the basis of ${\bf X}(\Omega)$ in ${\bf L}^{3+\delta}$. The proof of Lemma \[RegDHarV\] is complete.

A discrete Sobolev embedding inequality for the Nédélec element space {#SobNed}
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\[DisDiv\][ *For any given ${\bf a}_h\in {\mathbb N}_{h}^{k}$, the unique function $\zeta_h\in {\mathbb V}_h^{k+1}$ satisfying $$(\zeta_h, \chi_h)=-({\bf a}_h,\nabla\chi_h) ,
\quad\forall\, \chi_h\in {\mathbb V}_h^{k+1} ,$$ is called the discrete divergence of ${\bf a}_h$, denoted by $\zeta_h:=\nabla_h^{\mathbb N}\cdot{\bf a}_h$. The 