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?
extension of the exterior derivative
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differential-geometry
manifolds
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2Can you extend the usual derivative from $C^\infty(\mathbb R)$ to $L^2(\mathbb R)$? – 2012-08-19
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0$d$ is continuous so I would define it as $d(f)=\lim d(\psi_n )$ where $\psi_n \rightarrow f$ , $f \in L^2$ $\psi_n in C^\infty$ – 2012-08-19
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1$d$ is continuous with respect to what topology on $C^\infty$? Why would the limit you are proposing to use as definition exist? – 2012-08-19
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0@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 $H^{1,2}$ – 2012-08-19
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0i was addressing *half* of the question... – 2012-08-19
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1[This answer](http://math.stackexchange.com/questions/178657/averaging-differential-forms/178894#178894) might be relevant. – 2012-08-19