On the Structural Properties of the Weight Space $L_{p(x),\omega}$ for $0< p(x)<1$

The main purpose of this paper is to study the weight space $L_{p(x),\omega}$ for $0< p(x)<\nobreak 1$ as well as the topology of this space. Embeddings between different Lebesgue spaces with variable exponent of summability are established. In particular, it is proved that the set of all linear continuous functionals over $L_{p(x),\omega}$ for $0< p(x)<\nobreak 1$ consists only of the zero functional.