Norm on $H^1/2(\partial\Omega)$ by first isomorphism theorem.

Question

I want to show that $\operatorname{ran}(T)$ equipped with \begin{equation*} \lVert v\rVert:=\inf\{u\in H^1(\Omega):Tu=v\} \end{equation*} is a Hilbert-Space. I have seen that the definition of the norm is motivated by the norm on quotient spaces. It was suggested to show that $\hat{T}:x+\ker(T)\mapsto T(x)$ is an isomorphism form $H^1(\Omega)/\ker(T)$ to $H^{1/2}(\partial\Omega)$, by applying the first Isomorphism Theorem. How do I apply the Theorem without knowing that $\operatorname{ran}(T)$ is closed. I know that $\hat{T}$ is linear, bijective and continuous, how do I continue?