The following is the characterization theorem for $H^{-1}(U):=(H_0^1(U))^*$ in Evans's Partial Differential Equations:
Here is my question:
The proof says that (iii) directly follows from (i). Would anybody elaborate why it is so?
- $U$ is an open subset of $\mathbb{R}^{n}$
- $H^{1}(U)$ is the Sobolev space of $L^{2}(U)$ functions with weak derivatives in $L^{2}(U)$
- $H_{0}^{1}(U)$ is the closure of the subspace $\mathcal{C}_{c}^{\infty}(U)$ of compactly supported smooth functions on $U$
- $H^{-1}(U)$ is the (continuous) dual of $H_{0}^{1}(U)$
It's really a definition following from the inclusion $L^2(U) \subset H^{-1}(U)$. Perhaps it is better to write out the inclusion in more detail:
$$L^2(U) \sim (L^2(U))^* \subset (H^1_0(U))^* =: H^{-1}(U).$$
The inclusion above is true because the inclusion $H^1_0(U) \subset L^2(U)$ is continuous, i.e., $\|u\|_{L^2(U)} \leq \|u\|_{H^1_0(U)}$. So when we write $v \in L^2(U) \subset H^{-1}(U)$, what we really mean is that we are associating $v$ with the bounded linear functional on $L^2(U)$ given by
$$u \mapsto (v,u)_{L^2(U)} \ \text{for } u \in L^2(U),$$
which is also a bounded linear functional on $H^1_0(U)$ given by the restriction
$$u \mapsto (v,u)_{L^2(U)} \ \text{for } u \in H^1_0(U).$$
This is why we can write $\langle v,u\rangle = (v,u)_{L^2(U)}$ when $v \in L^2(U) \subset H^{-1}(U)$.
This is at least the canonical way to embed $L^2(U) \subset H^{-1}(U)$. We could, for instance, define $\langle v,u\rangle := 2(v,u)_{L^2(U)}$ for $v \in L^2(U)$ and $u \in H^1_0(U)$. This defines a bounded linear functional on $H^1_0(U)$, but is not canonical in the sense I described above.