Let $\Omega$ be a bounded open subset of $\mathbb{R}^3$ and $n$ the outward-pointing unit normal to the boundary of $\Omega$. We know that the space $(L^2(\Omega))^3$ of square integrable functions admits the following orthonormal decomposition: $$ (L^2(\Omega))^3=\nabla H_0^1(\Omega) \oplus H_0(div\,0,\Omega) \oplus \left(\nabla H^1(\Omega) \cap H(div 0,\Omega)\right), $$ where
$H^1(\Omega)$ and $H_0^1(\Omega)$ the the usual Sobolev spaces,
$H(div 0,\Omega):=\{u\in (L^2(\Omega))^3|div(u)=0\}$
and $H_0(div 0,\Omega):=\{u\in (L^2(\Omega))^3|div(u)=0, n\cdot u=0\}$
We want to construct a bounded real value operator defined on $(L^2(\Omega))^3$ such that its restriction on the subspace $\nabla H_0^1(\Omega)$ takes the value 1 and the other one on the subspace $W$ takes the value 0. Thanks.