Definition of $H^{-1}$ space in Evans' PDE book

Any Hilbert space is of course self dual. But this is only true if you use the inner product as the pairing. Note that $g\in H^{-1}$ acts on $f\in H^1_0$ (formally) by $g(f)=\int_U gf$, where no derivatives appear. Often the dual space of a function space is considered to act by "a simple integral", which may not be the inner product (if we are not in $L^2$).

Let me try to make this a bit more concrete. Let us write $\langle f,g\rangle=\int_Ufg$ and $(f,g)=\langle f,g\rangle+\langle \nabla f,\nabla g\rangle$. The second one is the inner product on $H^1_0$. Then $H^1_0\ni f\mapsto(f,\cdot)\in (H^1_0)'$ is surjective (as always in a Hilbert space), but $H^1_0\ni f\mapsto\langle f,\cdot\rangle\in (H^1_0)'$ is not.