In a paper, I've read the following thing. Here $\Omega$ is a smooth domain
From the standard trace theorem we know there exists a bounded linear operator $$\gamma: H^1(\Omega) \rightarrow H^{\frac{1}{2}}(\partial \Omega)$$ such that $\gamma_0 u=u_{|\partial W}$ for $u$ smooth and that satisfies $$||\gamma_0 u||_{H^{\frac{1}{2}}} \geq C ||u||_{H^1} \qquad \forall u\in H^1(\Omega) \setminus \ker(\gamma_0)$$
I don't know how to show the last equality.
If I can prove that $\gamma_0$ has closed range, then I have that it's bounded from below ( by a standard result coming from Open Mapping Theorem). However, I don't know how to prove it. Any hint is highly appreciated!