Embedding of $\mathcal{D}(\Omega)$ in $W^{k,p}_0(\Omega)$

Question

$W^{k,p}_0(\Omega)$ is defined as the closure of the set of all $C_c^{\infty}(\Omega)$ under the topology generated by the norm $W^{k,p}(\Omega)$. So clearly the identity map from $\mathcal{D}(\Omega) \rightarrow W^{k,p}_0(\Omega)$ defines an injection. Is this injection continuous?

If so how to prove it.

P.S:

1. $\mathcal{D}(\Omega)=(C_c^{\infty}(\Omega),\tau_{LF})$, i.e., $C_c^{\infty}(\Omega)$ endowed with its canonical LF topology.

2. $W^{k,p}_0(\Omega)=\big(\mathcal{D}(\Omega), \|\cdot\|_{W^{k,p}({\Omega})}\big)$, i.e., $C_c^{\infty}(\Omega)$ endowed with the topology generated by $W^{k,p}{\Omega}$ norm.

A clean proof will be greatly appreciated. Thanks in advance.

Answer 1

Enough to prove that if a sequence $\phi_n \rightarrow 0$ in $ \mathcal{D}(\Omega)$ then $\phi_n \rightarrow 0$ in $W^{k,p}$ norm.

Since $\phi_n \rightarrow 0$ in $\mathcal{D}(\Omega)$ there exists a compact set $K \subset \Omega$ such that $\operatorname{supp}(\phi_n) \subset K$ and for every multi index $\alpha$ $\|D^{\alpha}\phi_n\|_{L^{\infty}(K)} \rightarrow 0$. Consequenlty

\begin{align} \|\phi_n\|_{W^{k,p}(\Omega)} := \sum_{|\alpha| \leq k} \|D^{\alpha}\phi_n\|_{L^p(K)} \leq \mu(K)^{1/q} \sum_{|\alpha| \leq k} \|D^{\alpha}\phi_n\|_{L^{\infty}(K)} \rightarrow 0. \end{align}