Let
- $d\in\mathbb N$
- $\Omega\subseteq\mathbb R^d$ be open
- $\mathcal D:=C_c^\infty(\Omega,\mathbb R^d)$
- $\langle\;\cdot\;,\;\cdot\;\rangle$ denote the inner product on $L^2(\Omega,\mathbb R^d)$ and $$H:=\overline{\mathcal D}^{\langle\;\cdot\;,\;\cdot\;\rangle_H}$$ with $$\langle\phi,\psi\rangle_H:=\langle\phi,\psi\rangle+\sum_{i=1}^d\langle\nabla\phi_i,\nabla\psi_i\rangle\;\;\;\text{for }\phi,\psi\in\mathcal D$$
- $\iota:\mathcal D\to H$ be the inclusion
I've seen people talking about the adjoint $Q:H'\to\mathcal D'$ of $\iota$. But is the notion of an adjoint even defined for $\iota$?
As far as I know the notion of an adjoint is only defined for (bounded or unbounded) linear operators between Banach spaces, but $\mathcal D$ (equipped with the usual topology) is only a locally convex topological space.
If we we forget about that, we could treat $\iota$ as a densely-defined unbounded operator and would obtain $$QF=F\iota=\left.F\right|_{\mathcal D}\;\;\;\text{for all }F\in H'\;.\tag 1$$
Since the mapping $$ \iota\colon\mathcal{D}\to H $$ is a continuous linear operator (which is the "generalization" of bounded operators to the setting of locally convex spaces), there is a (continuous) transpose $$ \iota^t \colon H' \to \mathcal{D}' $$ satisfying $$ \langle \varphi, \iota^t(T)\rangle = \langle \iota(\varphi), T\rangle $$ for all $\varphi\in\mathcal{D}$ and all $T\in H'$, where $$ \langle \varphi, S\rangle := S(\varphi) $$ is the duality mapping between $\mathcal{D}$ and $\mathcal{D}'$.