The locally-convex space of *essentially compactly-supported $p$-integrable "functions"* $\operatorname{L}_{comp}^p(\mathbb{R}^d,\mathbb{R})$ is the define as the set
$$
\bigcup_{n \in \mathbb{N}} \left\{
f \in L^p(\mathbb{R}^d,\mathbb{R}):\, \operatorname{ess-supp}(f)\subseteq [-n,n]^d
\right\},
$$
topologized with the [injective limit topology][1] in the category of [LCSs][2] with continuous lienar maps as morphisms, where the colimit is taken over the [injective system][3] $\left\{L^p(\mathbb{R}^d,\mathbb{R}):\, \operatorname{ess-supp}(f)\subseteq [-n,n]^d\right\}_{n \in \mathbb{N}}$.  

Fix a $k \in \mathbb{N}$, $1\leq p<\infty$ and let $W^{k,p}_{comp}(\mathbb{R}^d):=W^{k,p}(\mathbb{R}^d) \cap \operatorname{L}_{comp}^p$.  How are the subspace topologies on $W^{k,p}_{comp}(\mathbb{R}^d)$ comparable?  Is the subspace topology induced by restricting the subspace topology on $L^p_{comp}(\mathbb{R}^d,\mathbb{R})$ to $W^{k,p}_{comp}(\mathbb{R}^d)$ at-least as fine than the one obtained by restricting the Sobolev topology $W^{k,p}$ to $W^{k,p}_{comp}(\mathbb{R}^d)$?


  [1]: https://en.wikipedia.org/wiki/Final_topology
  [2]: https://ncatlab.org/nlab/show/locally+convex+topological+vector+space
  [3]: https://en.wikipedia.org/wiki/Direct_limit