On the Spaces LpŽ x.Ž . and W m, pŽ x.Ž .1

In this paper we present some basic results on the generalized Lebesgue spaces L pŽ x .Ž . and generalized Lebesgue Sobolev spaces W m, pŽ x .Ž .. These results provide the necessary framework for the study of variational problems and elliptic equations with non-standard pŽx.-growth conditions.