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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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    [This answer](http://math.stackexchange.com/questions/178657/averaging-differential-forms/178894#178894) might be relevant.2012-08-19

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