Geometric Properties Of Vector-valued Bochner-Lebesgue Spaces With Variable Exponent

In this dissertation, first, the vector-valued Bochner-Lebesgue and Sobolev spaces with variable exponent are introduced. Then the properties:the com-pletion, the dual space, the reflexivity, the uniformly convexity and uniformly smoothness, of these new spaces are obtained. Those are the generalization of scalar valued Lebesgue and Sobolev spaces with variable exponent.In Chapter1, there are the history and the survey of the recent devel-opments of function spaces with variable exponent, the main results of this dissertation.In Chapter2,first, we prove that LP(·)(A.E) is complete. Secondly, we consider the dual of LP(·)(A,E) and obtain that LP(·)(A,E*) is isomorphic to (LP(·)(A, E))*when E*has the Radon-Nikodym property. Thirdly, by these results, the reflexivity, uniformly convexity and uniformly smoothness of these vector-valued Bochner-Lebesgue spaces with variable exponent are given. Fi-nally, the separability, reflexivity, and uniformly convexity of vector-valued Bochner-Sobolev spaces with variable exponent are obtained.