The locally convex space of *essentially compactly-supported $p$-integrable "functions"* $\operatorname{L}\_{\mathrm{comp}}^p(\mathbb{R}^d,\mathbb{R})$ is defined 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](https://en.wikipedia.org/wiki/Final_topology) in the category of [LCSs](https://ncatlab.org/nlab/show/locally+convex+topological+vector+space) with continuous linear maps as morphisms, where the colimit is taken over the [injective system](https://en.wikipedia.org/wiki/Direct_limit) $$\left\{L^p(\mathbb{R}^d,\mathbb{R}):\, \operatorname{ess-supp}(f)\subseteq [-n,n]^d\right\}\_{n \in \mathbb{N}}.$$ 


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



Sobolev topology on essentially compactly supported Sobolev-"functions"
To make my comment an answer: No. To see this, fix a cube $K$ and look at $L^p\_K=\{f\in L^p: \text{ ess-supp}(f) \subseteq K\}$. On this space, the $L^p\_{comp}$-topology coincides with the $L^p$-topology (this follows from the stricness of the inductive limit) and the $W^{k,p}$-topology is strictly finer on this subspace.


