Let $\Omega\in \mathbb{R}^d$ be a bounded simply connected domain. Suppose $\partial\Omega$ is split into disjoint $\partial\Omega_N$ and $\partial\Omega_D$ with $\partial\Omega = \overline{\partial\Omega_N} \cup \overline{\partial\Omega_D}$. Denote by $H_{00}^{1/2}(\partial\Omega_N)$ as
$$ H_{00}^{1/2}(\partial\Omega) = \{v\in H^{1/2}(\partial\Omega)\colon \exists V\in H^{1}(\Omega)\quad V\big|_{\partial\Omega_N} = v\, \mathrm{and} \quad V\big|_{\partial\Omega_D} = 0\} $$
Let $H^{-1/2}(\partial\Omega_N)$ denote the dual space of $H^{1/2}_{00}(\partial\Omega)$.
Suppose $\Gamma$ is a simply connected proper subset of $\partial\Omega_N$, and $f\colon\Gamma \to \mathbb{R}$ is a polynomial. Is there any sense in which $f$ is as an element of dual space of $H^{-1/2}(\partial\Omega_N)$, (the bi-dual of $H^{1/2}_{00}(\partial\Omega)$)?
Since $H^{1/2}_{00}(∂Ω)$ is a Hilbert space, it is equal to its own bi-dual. And piecewise constants are not in $H^{1/2}$.