If the space $H^1_0(\Omega)$ is equipped with the norm
$$\|v\|^2:=\epsilon\|v\|_{2}^2+\|\nabla v\|_{2}^2$$
then, my first question is
If $f\in H^{-1}(\Omega)$ then does
$$\sup_{v\in H^1_0(\Omega)}\dfrac{\langle f,v \rangle}{\|v\|}= \|f\|_{H^{-1}(\Omega)}$$
or
$$\sup_{v\in H^1_0(\Omega)}\dfrac{\langle f,v \rangle}{\|v\|}= \|\epsilon^{-\frac{1}{2}}f\|_{H^{-1}(\Omega)}$$
?
My second question is
If $g \in H^1_0(\Omega)$, then can I say that
$$\|g\|_{H^{-1}(\Omega)}=\|g\|_{L^{2}(\Omega)} $$
?
Many thanks for your help.