This chapter is devoted to developing function spaces that are used in the variational formulation of differential equations. We begin with a review of Lebesgue integration theory, upon which our notion of “variational” or “weak” derivative rests. Functions with such “generalized” derivatives make up the spaces commonly referred to as Sobolev spaces. We develop only a small fraction of the known theory for these spaces—just enough to establish a foundation for the finite element method.


Banach Space Sobolev Space Dual Space Lebesgue Space Lipschitz Domain 


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© Springer Science+Business Media, LLC 2008

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