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 \(X^p\)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 \(X^p\)space with respect to the Banach function space X is the corresponding grand \(X^p\)space.

中文翻译：

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大和小$$ X ^ p $$ X p空间和广义对偶

在本文中，对于有限度量空间上的一般Banach函数空间，我们扩展了大Lebesgue空间和小Lebesgue空间的构造。我们称这些空间为大和小\（X ^ p \）空间。我们证明了这些空间的几个基本属性的结果，即对偶性，重排不变性以及从原始空间X转移到相应的大空间和小空间的其他属性。特别地，关于对偶性，我们证明了小\（X ^ p \）空间相对于Banach函数空间X的广义关联空间是相应的大\（X ^ p \）空间。