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There exists an embedding relation “$\subset$” or “$\supset$” between $C^{\infty}_c (\mathbb{R}^n)$ and $H^1(\mathbb{R}^n)$?

Could anyone give me an example? Thank you in advance!

C. Bishop
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    PS if theres anything you want me to expand on with proofs or pictures etc, I'll be glad to. Also: your previous question here https://math.stackexchange.com/questions/3601991/is-this-dense-embedding-true with basic modifications (as indicated by a comment) here gives the density of $C^1_c$ in $H^1(\mathbb R^N)$. $C^\infty_c$ is of course dense in $C^1_c$. And your question from > 1 year ago seems like it covers this question and more https://math.stackexchange.com/questions/3168100/f-in-c-0-infty-rightarrow-f-in-lm – Calvin Khor May 11 '20 at 02:49
  • It is enough for now, thank You so much @Calvin Khor! – C. Bishop May 11 '20 at 07:18
  • You're welcome :) – Calvin Khor May 11 '20 at 07:19

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A smooth compactly supported function has weak derivatives of all order that are in every $L^p$ space. In particular, $$ C^\infty_c (\mathbb R^n)\subset H^1(\mathbb R^n).$$ In fact more is true, $C^\infty_c(\mathbb R^n)$ is dense in $H^1(\mathbb R^n)$. A $C^1_c$ function that is not smooth shows that $H^1\neq C^\infty_c$.

"bump functions" are your standard example of a function in $C^\infty_c$, and there's lots of discussion on them already on MSE. For example : Where can we find examples of smooth functions with compact support? Is there a book?. This example is related to the smooth but non-analytic function in this wikipedia page.

These examples are for 1 variable but it is trivial to modify them for higher dimensions by e.g. forcing them to be radially symmetric, or multiplying toegether one for each coordinate.

Calvin Khor
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