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Sobolev Topology on Essentially Compactly Supported Sobolev-"Functions"

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 in the category of LCSs with continuous lienar maps as morphisms, where the colimit is taken over the injective system $\left\{L^p(\mathbb{R}^d,\mathbb{R}):\, \operatorname{ess-supp}(f)\subseteq [-n,n]^d\right\}_{n \in \mathbb{N}}$.

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 $W^{k,p}_{comp}(\mathbb{R}^d)$?

ABIM
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