# Sobolev topology on essentially compactly supported Sobolev-"functions"

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 in the category of LCSs with continuous linear 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 $$W^{k,p}_{\mathrm{comp}}(\mathbb{R}^d):=W^{k,p}(\mathbb{R}^d) \cap \operatorname{L}_{\mathrm{comp}}^p$$. 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 $$W^{k,p}_{\mathrm{comp}}(\mathbb{R}^d)$$?

• Why should this be true? The $L^p_{comp}$-topology gives nothing for the distributional derivatives! You probably know that because of the stricness of the inductive system the relative topology of $L^p_{comp}$ on the steps is just the $L^p$-topology. Mar 25 '20 at 15:05
• But in this post: mathoverflow.net/questions/347318/… don't we find that the topology on $L^p_{comp}$ is strictly finer? Mar 25 '20 at 15:15
• Indeed, the $L^p_{comp}$ topology is stricly finer than the $L^p$-topology, but on the steps they coincide. There is no contradiction. Mar 25 '20 at 15:21
• Oh my brain filtered the word "steps" sorry. I've never seen this terminology, you mean the finite "sub-colimits?". In that case definitely. Mar 25 '20 at 15:25

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.