Let $\Omega$ be the open unit ball in $\mathbb{R}^n$, and $\Gamma := \Omega \cap \{x_n=0\}$.
Let $\Omega_1 = \{ x \in \Omega: x_n > 0 \}$ and $\Omega_2 = \{ x \in \Omega: x_n < 0 \}$.
Define $H$ to be the completion of $C^0(\overline{\Omega}) \cap H^{1}(\overline{\Omega})$ with the norm \begin{equation} \|u\|_H := \left( \int_\Omega (u^2 + |\nabla u|^2) dx + \int_\Gamma |\nabla'u|^2 dx \right)^{1/2}, \end{equation} where $\nabla'u$ is the tangential derivative of $u$.
1) When I want to define a Hilbert space, my professor told me to define it as a completion of another space. Why is it so? I think a well-defined inner product on a complete metric space should be enough.
2) I think $H$ is a Hilbert space, but is this space well-defined, or do I need "the completion of $C^1(\overline{\Omega})$"?
3) About traces, let $T_i$ be the trace on $\Gamma$ from $\Omega_i$ for $i \in \{1,2\}$. Then $T_i: H^s(\Omega_i) \to H^{s-1/2}(\Gamma)$, $s > \frac{1}{2}$ is bounded. Can we say that if $u \in H$, $T_1 u = T_2 u$ on $\Gamma$?
4) Can we also have $T_1 \partial_r u = T_2 \partial_r u$ for $1 \le r \le n-1$?
PS: Everything is real-valued.
Thank you.