Some notes about Orlicz spaces related to a Banach function space

Orlicz spaces, as a very important generalization of Lebesgue spaces, were proposed more than 100 years ago and were first introduced by Z. W. Birnbaum and W. Orlicz. After that, another generalization of Lebesgue spaces with symbols X^p instead of L^p was introduced in which X is a Banach function space. In this article, another generalization of Lebesgue spaces is investigated, which includes both previous generalizations. Although this generalization and preliminary studies of it were published in a scientific report in 1988, effective research on this generalization was done 15 years ago. Due to the fact that in this generalization a convex and Young function Φ is placed instead of function |.|^p and a Banach function space X is placed instead of space L^1, in this article, taking into account the conditions of the function Φ we will check properties of generalized Orlicz space X^Φ.