Consider the negative Sobolev space $W^{-k,p}(\Omega)=(W^{k,q}_0(\Omega))^*$ for $k\in \mathbb{N}$ and $1/p+1/q=1$.
Since $C_c^{\infty}(\Omega) \subset W^{k,p}_0(\Omega),$ clearly an element of $W^{-k,p}(\Omega)$ maps $C_c^{\infty}(\Omega)$ to $\mathbb{R}$. How to prove that it is a disbribution. In otherwords, does continuity wrt $W^{k,q}$ norm imply continuity in the inductive limit topology on $\mathcal{D}(\Omega)$?
A clean proof is greatly appreciated.
Let $u \in W^{-k,p}(\Omega)$ then for every compact set $K$ and $\phi \in \mathcal{D}(\Omega)$ with $\operatorname{supp}(\phi) \subset K$ we have: \begin{align} \langle u,\phi\rangle& \leq C \left(\sum_{|\alpha|\leq k} \|D^{\alpha}\phi\|_{L^q(K)} \right)\\& \leq C \mu(K)^{1/q} \sum_{|\alpha| \leq k } \|D^{\alpha} \phi\|_{L^{\infty}(K)} \\&:=C_K \sum_{|\alpha| \leq k } \|D^{\alpha} \phi\|_{L^{\infty}(K)} , \end{align} which implies that $u$ is a distribution of order less than or equals $k$.