Norm equivalence between Sobolev space and L_2

[kisengue]

Hello! I've found this paper, wherein page 33 states that the reverse Poincaré inequality gives

$$\forall v \in H^1_0(\Omega) , \|v\|_{L^2(\Omega)} \leq C(\Omega) \|\nabla v\|_{L^2(\Omega)}$$

This I can follow - it gives a norm equivalence between the norm of a vector and the gradient of its gradient (clumsily expressed, I know). However, just a little bit later the paper states that

$$\|u\|_{H_0^1(\Omega)} = \|\nabla u\|_{L^2(\Omega)}$$

That is, those two norms are equivalent. This is what I don't understand - I don't understand the jump from the L^2-norms in the first statement to the Sobolev norm in the second. Any help in understanding this would be helpful.

 

[bigfooted]

I think L2 is also in a Hilbert space. And if you take the gradient of a function, the result resides in a lower Hilbert Space.
So I think the problem reduces to finding out which H corresponds to L2.