Riesz map between Sobolev space and its dual

Question

I'm asked to find the Riesz's map:

$$R:H^1_0(\Omega) \rightarrow H^{-1}(\Omega) $$ $$R: u \mapsto F_u \quad \text{s.t.} \quad <F_u,v>_*= (u_F,v)_{H_0^1} \ \ \forall v \in H_0^1 $$

I chose $(u,v)_{H_0^1} = (\nabla u, \nabla v)_{L^2}$

Here's my thoughts: I need to find $F_u$:

$$(u,v)_{H_0^1} = <F_u,v>_* \forall v \in H_0^1 $$ $$-\Delta u =F_u$$

And this would be my solution, $R=-\Delta$

However, I found online that $R=I-\Delta$ ($I$ being the identity)

Can someone explain to me why my reasoning is wrong and how to find such a result?

Answer 1

If you use $(u, v)_{H_0^1} = (\nabla u, \nabla v)_{L^2}$, you get $R = -\Delta$, as you have shown.

And there is another scalar product on $H_0^1$, such that $R = I - \Delta$. Which one?