Grand and small $$X^p$$spaces and generalized duality

Abstract

In this paper we extend the construction of Grand and Small Lebesgue spaces for the case of general Banach function spaces on finite measure space. We call these spaces the grand and the small $$X^p$$\n spaces. We prove results on several fundamental properties of these spaces, namely, duality, rearrangement invariant and the other properties that are transferred from the original space X to the corresponding grand and small spaces. In particular, on duality, we show that the generalized associate space of the small $$X^p$$\n space with respect to the Banach function space X is the corresponding grand $$X^p$$\n space.