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Understanding denseness of $C^\infty$ in $L^p$ space.
I am looking for a proof that shows $C^\infty(\Omega)$ is dense in $L^p(\Omega)$ . Any hints would be appreciated. Where $\Omega\subset \mathbb R^n$.
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What is $\Omega$? – Alex Becker May 20 '12 at 09:27
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Its a bounded, open connected subset of $\mathbb R^n$ – Theorem May 20 '12 at 09:30
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- Show the continuous functions with compact support are dense. 2. Use mollifiers to approximate continuous functions uniformly by smooth functions.
– t.b. May 20 '12 at 09:39 -
@t.b. can you explain me why do we have to take continuous functions to have compact support ? – Theorem May 20 '12 at 09:44
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1@Ananda I would think that's because otherwise you're not sure whether the convolution exists. But if $f,g$ both have compact support then $f \ast g$ exists. See here. – Rudy the Reindeer May 20 '12 at 09:51
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1This is related. – Rudy the Reindeer May 20 '12 at 09:55
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Aren't polynomials dense in $L^p(\Omega)$? – dtldarek May 20 '12 at 16:01
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1Ok, I missed that your $\Omega$ is open. But again, this might be helpful ;-) – dtldarek May 20 '12 at 16:09