Can I equip the space $H^{1/2}(\partial \Omega)$ with a $L^2( \partial \Omega)$ inner product?

Question

Since the space $H^{1/2}(\partial \Omega)$ with its norm $\Vert u \Vert_{H^{1/2}(\partial \Omega)} = \inf_{v \in H^1(\Omega), v\vert_{\partial \Omega} = u} \Vert v \Vert_{H^1(\Omega)}$ gets unhandy in numerical computations, I would like to know, if I can equip this space with the inner product of the space $L^2( \partial \Omega)$.

2) side question: Is there any chance, I can get $\Vert u \Vert_{H^{1/2}(\partial \Omega)} \leq C \Vert u \Vert_{L^2(\partial \Omega)}$?

Answer 1

For numerical computation, maybe you can try another definition of $H^{\frac 1 2}(\partial\Omega)$, see https://math.stackexchange.com/q/2311227 .