If and is a domain (in a -dimensional real space with easy generalization to manifolds), one first considers the Lebesgue spaces (wikipedia) of (equivalence classes of) measurable (complex- or real-valued) functions whose (absolute values of) -th powers are Lebesgue integrable; i.e. whose norm
is finite. For , one looks at the essential supremum norm instead.
For , and the Sobolev space or is the space of measurable functions on such that its generalized partial derivatives (e.g. in the sense of generalized functions) for all multiindices with are in . The most important case is the case of the Sobolev spaces . Sobolev spaces are particularly important in the theory of partial differential equations.
L. C. Evans, Partial Differential Equations, Amer. Math. Soc. 1998.
R.A. Adams, Sobolev spaces, Acad. Press 1975.
wikipedia: Sobolev space.
Springer Encyclopaedia of Mathematics: Sobolev space
G. Wilkin, Sobolev spaces on Euclidean space, Liber Mathematicae 2011, link