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I'm reading by myself the PDE's book by Lawrence Evans and he denoted the dual of $H^1_0(U)$, which is the closure of $\mathcal{C}^{\infty}_c(U)$ in $W^{1,2}(U)$ ($H^1_0(U) = W^{1,2}_0(U)$ ), by $H^{-1}(U)$. In what follows, Evans states

$\textit{Note very carefully that we do not identify the space}$ $H^1_0(U)$ $\textit{with its dual}$. Instead, as we will see in a moment, we have

$$H^1_0(U) \subset L^2(U) \subset H^{-1}(U). (*)$$

I can see that $H^1_0(U) \subset L^2(U)$ since $H^1_0(U) = \overline{\mathcal{C}^{\infty}_c(U)} \subset W^{1,2}(U) \subset L^2(U)$, but I can't see why $L^2(U) \subset H^{-1}(U)$ since $H^{-1}(U)$ is the space of bounded linear functions and I don't know if all $f \in L^2(U)$ is a linear function. I understand $(*)$ literally as a space contained in another space, but the author of this OP led me to think that $(*)$ maybe can be true doing identifications, as the author of the topic did, so my question is if $(*)$ is valid doing identifications or not. If $(*)$ is valid by continence of spaces, then why $L^2(U) \subset H^{-1}(U)$?

Thanks in advance!

Community
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George
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1 Answers1

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You should make some kind of identification, because the objects you are taking into account are logically different: $L^2$ is the space of equivalence classes of certain functions from $\mathbb{R}$ to $\mathbb{R}$, while $H^{-1}$ is a space of functions from $H^1_0$ to $\mathbb{R}$. The point is that there is a natural identification of a function $f\in L^2$ with a dual function $$ H^1_0 \to \mathbb{R},\ g \mapsto \int_\Omega fg = (f,g)_{L^2} $$ This identification is natural as it is induced by the scalar product on $L^2$, which is "part of" the scalar product on $H^1$. Notice that the idenfication we are considering is $f \eqsim (f,\cdot)$, which is indeed in $H^{-1}$ since $$ (f,g) = \int_\Omega fg \leq \|f\|_2 \|g\|_2 \leq \|f\|_2 \|g\|_{H^1_0}. $$

Tommaso Seneci
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