Distributions and duality in Sobolev spaces

The dual space of a Sobolev space is not only composed of functions (defined almost everywhere), but it also contains more sophisticated objects called distributions which are defined by their action on smooth functions with compact support. Dual Sobolev spaces are useful to handle singularities on the right-hand side of PDEs. They are also useful to give a meaning to the tangential and the normal traces of vector-valued fields that are not smooth enough to admit traces as members of a suitable Sobolev space. The extension of the notions of tangential and normal traces is done in this case by invoking integration by parts formulas involving the curl or the divergence operators.