Can we talk about the adjoint of a linear operator defined on a distribution space?

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 $\mathcal{D}'$.
• Why is $\iota$ continuous? Is it cause $$\left\|\iota\phi\right\|_H\stackrel{\text{def}}=\left(\left\|\phi\right\|^2+ \sum_{i=1}^d\left\|\nabla\phi_i\right\|^2\right)^{\frac 12}\le\lambda(K)^{\frac 12}\left(\sup_K|\phi|^2+\sum_{i=1}^d\sup_K|\nabla\phi_i|^2\right)^{\frac 12}\stackrel{\text{def}}=\lambda(K)^{\frac 12}\left\|\phi\right\|_{C^1(K)}$$ for all compact $K\subseteq\Omega$ and $\phi\in\mathcal D$ with $\operatorname{supp}\phi\subseteq K$. – 0xbadf00d Jul 8 '16 at 14:58
• @0xbadf00d: The continuity indeed follows from the above estimate. Note that this does not mean that $\|\iota\|\in\mathcal{D}'$ since it is not linear – Christian Jul 8 '16 at 15:09
• Sorry, I was too hasty. $\left\|\iota\right\|_H\in\mathcal D'$ is obviously wrong, cause $\left\|\;\cdot\;\right\|_H$ is only sublinear. How does this transpose differ from the notion of an adjoint of an operator between Banach spaces? Clearly, $\mathcal D$ is not a Banach space. But I don't see anything in the linked definition of an adjoint what wouldn't make sense for an operator $\mathcal D\to H$ too. – 0xbadf00d Jul 8 '16 at 15:09
• So, $\iota^t$ is exactly the continuous linear map $Q$ from the question, right? – 0xbadf00d Jul 8 '16 at 15:25