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A Sobolev space is a vector space of functions equipped with a norm that is a combination of Lp-norms of the function itself and its derivatives up to a given order.
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Boundary Value Problem in the space of Distributions
I think what you are looking for might be explained in Section 18.3 and Appendix B.2 in Lars Hörmander, The Analysis of Linear Partial Differential Operators III, Springer-Verlag.
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