On a Closed Subspace of L^(p(.)(Ω))

Abstract

In this study, we first give a description of L^(p(.)(Ω)) spaces. These spaces are an important generalization of classical Lebesgue spaces. We mention their various applications in engineering and physics fields. Thereafter, as it is naturally, one of the main task in L^(p(.)(Ω)) spaces is to generalize known properties classical Lebesgue spaces L^p(Ω)) to L^(p(.)(Ω)) spaces. Provided that measure of the set Ω is finite, we extend a theorem which about a closed subspace of space, from constant exponent to variable exponent. Our proof method based on embedding between L^(p(.)(Ω)) - L^p(Ω)) spaces and the proof of constant case. The essence of the method is to take advantage of properties of Hilbert space L^2(Ω)), and also based on the use of the closed graph theorem and finite measure of the set Ω.