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Let $\mathcal S (\Bbb R^n)$ be the Schwartz class and $W^{s,2}(\Bbb R^n)$ be the Sobolev space($s=0,1,\cdots$). In fact I know that $C_c^\infty(\Bbb R^n)$ is dense in $W^{s,2} (\Bbb R^n)$ and $C_c^\infty (\Bbb R^n) \subset \mathcal S(\Bbb R^n)$. But how can I prove that $\mathcal S(\Bbb R^n)$ is dense in $W^{s,2} (\Bbb R^n)$ directly?

$C_c^\infty$ : $C^\infty$ with compact support.

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I dont understand you question, because proving that $C_c^\infty(\mathbb{R}^n))$ is dense $W^{s,2}$ is a direct proof that Schwartz spaces is dense in $W^{s,2}$. Can you explain it better please? – Tomás Jan 30 '13 at 10:56

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