模糊赋范Riesz空间上序列基本性质的研究

Research on the Basic Property of Sequence on Fuzzy Normed Riesz Space

摘要: 模糊赋范Riesz空间是一个既具有Riesz空间的序结构又具有模糊赋范空间的模糊范数结构的线性空间，它将Riesz空间(也称为向量格或向量格子)与模糊赋范空间的概念结合起来。在模糊赋范Riesz空间中，研究序列的收敛性、有界性和完备性是十分重要的。本文通过对序列性质的研究，给出了模糊赋范Riesz空间上模糊序闭集的概念，讨论了模糊Banach格中模糊范收敛和一致收敛的关系，并用序列的模糊范收敛和完备性来讨论模糊赋范Riesz空间上的相关性质，最后讨论了模糊Banach格中模糊序连续范数的充要条件，丰富和推广了已有结论。

Abstract: A fuzzy normed Riesz space is a linear space that has both the order structure of a Riesz space and the fuzzy norm structure of a fuzzy normed space, which combines the concepts of a Riesz space (also known as a vector lattice or a vector lattice module) and a fuzzy endowed Riesz space. In a fuzzy normed Riesz space, the study of convergence, boundedness, and completeness of sequences is very important. In this paper, by studying the properties of sequences, the concept of a fuzzy order closed set in a fuzzy normed Riesz space is given, followed by a discussion of the relationship between fuzzy norm convergence and uniform convergence in a fuzzy Banach lattice and the use of sequence fuzzy norm convergence and completeness to discuss the differences and connections between a fuzzy normed Riesz space and a fuzzy Banach lattice, finally, the sufficient and necessary conditions for the fuzzy order continuous norm in the fuzzy Banach lattice are discussed, enriching and extending existing results.