In this chapter, we present the variable exponent Lebesgue spaces defined by Orlicz1) in 1931. Although the variable exponent Lebesgue spaces are introduced in 1931, it began to be actively studied in 1990s. We especially used the resources by Diening, Harjulehto, Hasto, Ruzicka 2011; Fan, Zhao 2001; Kovacik, Rakosnik 1991; Cruz-Uribe, Fiorenza 2013; Radulescu and Repovs 2015 for this section. The notion variable exponent Lebesgue and Sobolev spaces is directly related to the classical Lebesgue and Sobolev spaces where the constant p is replaced with the function p(.) which may depend on a variable. Further properties of these spaces are introduced and analyzed in that chapter.
Keywords: Variable exponent, exponent function, locally log-Holder continuous, variable exponent Lebesgue space, Luxemburg norm, Holder inequality, variable exponent Sobolev space, reflexive space, unit ball property, Lipschitz-continuous function.