Canonical edge moments are not bounded on H(curl)

By [F4] with $r=1\mathrm{/}2$ and $R=1$, choose $\eta \in C_c^{\mathrm{\infty }}({\mathbb{R}}^3)$ with $\eta (0)=1$ and support in the unit ball. Choose $\epsilon >0$ small enough that $b\mathrm{\notin }a+\epsilon \mathrm{supp}\eta$. Define ${\psi }_{\epsilon }(x)=\eta ((x-a)\mathrm{/}\epsilon )$ on $\mathrm{\Omega }$ and $u_{\epsilon }=\mathrm{\nabla }{\psi }_{\epsilon }$. Then $\mathrm{curl}{u}_{\epsilon }=0$ distributionally because distributional mixed derivatives commute, so [F1] gives $\mathrm{\parallel }{u}_{\epsilon }{\mathrm{\parallel }}_{H(\mathrm{curl})}=\mathrm{\parallel }{u}_{\epsilon }{\mathrm{\parallel }}_2$. A change of variables on ${\mathbb{R}}^3$ bounds $\mathrm{\parallel }{u}_{\epsilon }{\mathrm{\parallel }}_2^2\le \epsilon \mathrm{\parallel }\mathrm{\nabla }\eta {\mathrm{\parallel }}_{L^2({\mathbb{R}}^3)}^2\to 0$.

By [F3] along $e$, ${\mathrm{\ell }}_e(u_{\epsilon })={\psi }_{\epsilon }(b)-{\psi }_{\epsilon }(a)=0-1=-1$ for every such $\epsilon$. If a bounded extension existed, [F2] would give $\mathrm{\mid }{\mathrm{\ell }}_e(u_{\epsilon })\mathrm{\mid }\le C\mathrm{\parallel }{u}_{\epsilon }{\mathrm{\parallel }}_{H(\mathrm{curl})}$, which tends to zero by step 1.1, contradicting the fixed value $1$. Thus the edge moment is not bounded on the energy space; canonical interpolation cannot be applied to arbitrary H(curl) fields by these moments alone.