We have the operator $u \mapsto (u\vert_{\partial \Omega}, \frac{\partial u}{\partial n}\vert_{\partial \Omega})$, where $u \in H^1(\Omega)$ (Hilbert space), $\partial \Omega$ is the boundary $\Omega$, $n$ is the outward unit normal vector to $\partial \Omega$, and $\Delta u = 0$.
From "Functional Analysis, Haim Brezis", I know that $(u\vert_{\partial \Omega}, \frac{\partial u}{\partial n}\vert_{\partial \Omega}) \in H^{\frac{1}{2}}(\partial \Omega)\times H^{-\frac{1}{2}}(\partial \Omega)$.
Is $H^{\frac{1}{2}}(\partial \Omega)\times H^{-\frac{1}{2}}(\partial \Omega)$ a Hilbert space?