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Is it true that since the smooth forms are dense in the $L^2$ and $H^{1,2}$ sections of forms, we can extend the exterior derivative $d$ and its adjoint $d^*$ to this spaces?

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Can you extend the usual derivative from $C^\infty(\mathbb R)$ to $L^2(\mathbb R)$? –  Mariano Suárez-Alvarez Aug 19 '12 at 18:59
    
$d$ is continuous so I would define it as $d(f)=\lim d(\psi_n )$ where $\psi_n \rightarrow f$ , $f \in L^2$ $\psi_n in C^\infty$ –  inquisitor Aug 19 '12 at 19:38
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$d$ is continuous with respect to what topology on $C^\infty$? Why would the limit you are proposing to use as definition exist? –  Mariano Suárez-Alvarez Aug 19 '12 at 19:40
    
@MarianoSuárez-Alvarez: $H^{1,2}$ is the space of functions with one square integrable derivative. Why do you unsettle the OP like that? As a side remark: your first comment, in my opinion, should read: can you extend the usual derivative to $H^{1,2}$ –  user20266 Aug 19 '12 at 20:07
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This answer might be relevant. –  user31373 Aug 19 '12 at 20:42

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