AnewdefinitionoflocallyconvexL-topologicalvectorspaces资源-CSDN文库

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在数学领域中，拓扑向量空间理论是泛函分析的重要组成部分，局部凸性是这个理论中的一个关键概念。在给定文件中，张化朋和方锦暄两位作者提出了局部凸L-拓扑向量空间的一个新定义，并研究了这个新定义与先前由严从华和方锦暄给出的定义之间的关系。L-拓扑向量空间是L-模糊拓扑向量空间的简称，它拓展了经典拓扑向量空间和Katsaras提出的I-拓扑向量空间的概念。这里的I指的是实数区间[0,1]。我们需要了解什么是拓扑向量空间。在拓扑向量空间的框架下，我们不仅研究向量加法和标量乘法，还要研究它们在拓扑结构下的连续性和相关性质。拓扑向量空间的一个重要子类是局部凸空间，这是因为在许多分析问题中，局部凸性是一个非常自然的假设，它可以确保许多重要的性质，比如Hahn-Banach定理在其中的成立。接下来，关于局部凸L-拓扑向量空间的新定义，这意味着在L-拓扑向量空间的背景下，我们能够讨论局部凸性这一结构。文件中提到了“广义L-fuzzy半范数”的概念，这显然是一个由L-fuzzy半范数概念推广而来的新概念。L-fuzzy半范数是一种度量，在L-拓扑向量空间中用来描述向量的“长度”或“大小”。广义L-fuzzy半范数的引入，为局部凸L-拓扑向量空间提供了新的度量方式，从而允许研究者用一套不同的数学工具来探讨这一空间的性质。张化朋和方锦暄的工作通过对广义L-fuzzy半范数的研究，提出了一种刻画局部凸L-拓扑向量空间的新方法。他们通过一族广义L-fuzzy半范数来表征这些空间，即通过这些半范数的结构来界定空间的局部凸性。这种表征方式可能与之前严从华和方锦暄的定义有着显著的区别，从而为局部凸L-拓扑向量空间的理论增添了新的视角和工具。在局部凸L-拓扑向量空间的研究中，Hausdorff分离性质、分子网络的收敛性以及L-模糊集的有界性是被研究者关注的几个重要方面。Hausdorff分离性质的讨论保证了空间中的点可以被不同的开集分离，这对于理解空间的拓扑结构至关重要。分子网络的收敛性则涉及到在空间中序列和网的极限行为，这对于分析空间中序列的性质至关重要。L-模糊集的有界性研究则直接关系到向量空间中的集合是否能够被控制在一定的规模之内。文章还提到了L-拓扑向量空间中的L-模糊概率半范数和L-模糊伪范数等概念，这些都是研究L-拓扑向量空间时不可或缺的工具。L-模糊概率半范数和L-模糊伪范数在描述空间的局部凸性和整体结构方面发挥着核心作用。张化朋和方锦暄的工作不仅为局部凸L-拓扑向量空间提供了一个新的定义，还通过广义L-fuzzy半范数的概念进一步丰富了L-拓扑向量空间的理论。他们的研究成果对于理解L-拓扑向量空间，尤其是在局部凸性方面，具有重要的数学意义，也有望在函数分析及其应用中发挥实际作用。

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A new deﬁnition of locally convex

L-topological vector spaces

∗

Hua-peng Zhang

†

, Jin-xuan Fang

Department of Mathematics, Nanjing Normal University, Nanjing, Jiangsu 210097

Abstract

In this paper, a new deﬁnition of locally convex L-topological vector spaces is given.

The relationship between this new deﬁnition and the previous deﬁnition of locally convex

L-topological vector spaces given by Yan and Fang [L-fuzzy locally convex topological

vector spaces, J. Fuzzy Math. 7 (3) (1999) 765−772] is investigated. Moreover, the

concept of generalized L-fuzzy semi-norm is introduced. By using a family of ge neralized L-

fuzzy semi-norms, a characterization of the new locally convex L-topological ve ctor spaces

is presented. Finally, as applications of this characterization, the Hausdorﬀ separation

property, conve rgence of molecule nets and boundedness of L-fuzzy sets in locally convex

L-topological vector spaces are studied.

Keywords: Analysis; L-topological vector spaces ; Locally convex L-topological vector

spaces; Generalized L-fuzzy semi-norms

1. Introduction

Fang and Yan [2] introduced the notion of L-fuzzy topological vector space in 1997.

According to the standardized terminology in [3], more accurately, it should be called

L-topological vector space (brieﬂy, L-tvs), which is the generalization of both the notion

of classical topological vector space and that of I-topological vector space introduced by

Katsaras [4], where I denotes the real unit interval [0, 1].

It is well-known that the theory of locally convex topological vector spaces plays a

quite important role in classical functional analysis and its applications. So, it is natural

∗

This work is supported by the National Natural Science Foundation of China (No. 10671094) and the

Specialized Research Fund for the Doctoral Program of Higher Education of China (No. 20060319001).

†

Corresponding author. E-mail address: huapengzhang@163.com.

http://www.paper.edu.cn

and necessary to ﬁnd the counterpart of locally convex topological vector s paces in the

research of L-tvs. In 1999, Yan and Fang [10] introduced the concept of locally convex

L-tvs by means of a prebase and characterized such space by a family of L-probabilistic

semi-norms. Moreover, Yan [9] als o characterized such space by a family of L-fuzzy pseudo-

norms. However, this deﬁnition of Yan and Fang only relates to a class of L-topological

vector spaces with a prebase, and it can not contain the deﬁnition of locally convex I-

topological vector space given by Katsaras [5] as a special case. In order to overcome these

limitations, in this paper we intend to redeﬁne and to study the locally convex L-tvs.

The present paper is organized as follows. In Section 2, we recall some basic concepts

and prove several lemmas to be needed in the sequel. In Section 3, we propose a new

deﬁnition of locally convex L-tvs, which is more reasonable than Yan and Fang’s deﬁnition.

Locally convex L-tvs deﬁned by Yan and Fang and locally convex I-tvs deﬁned by Katsaras

are both special cases of the new deﬁnition. In Section 4, we introduce the notion of

generalized L-fuzzy semi-norm and prove that every locally convex L-tvs deﬁned by our

way can be characterized by a family of generalized L-fuzzy semi-norms. In Section 5, as

applications of the results obtained in Section 4, we characterize the Hausdorﬀ separation

property, conve rgence of molecule nets and boundedness of L-fuzzy sets in locally convex

L-tvs by means of the family of generalized L-fuzzy semi-norms.

2. Basic concepts and lemmas

Throughout this paper, L denotes a complete and completely distributive lattice

equipped w ith an order-reversing involution

: L → L. 0 and 1 are its bottom and

top elements, respectively. M(L) denotes the set of all non-zero union-irreducible ele-

ments in L. The elements of M(L) are also called molecules [6] in L. For each λ ∈ L,

β(λ) denotes the standard minimal family of λ. L

denotes the family of all L-fuzzy sets

on X. An L-fuzzy set which takes the constant value λ ∈ L on X is denoted by λ

. An

L-fuzzy set on X is called an L-fuzzy point if it takes the value 0 for all y ∈ X except

one, say, x ∈ X. If its value at x is λ ∈ L \ {0}, we denote this L-fuzzy point by x

. An

L-topology δ on X is called stratiﬁed, if it contains all c onstant L-fuzzy sets on X. We

always assume that the L-topologies referred to in the present paper are all stratiﬁed and

the lattice L is regular (i.e., the intersection of each pair of non-zero elements in L is not

zero, which is equivalent to 1 ∈ M(L)). For other symbols which are not mentioned, we

refer to [2, 6].

Deﬁnition 2.1 (Liu and Luo [6], Wang [7] ). Let (L

, δ) be an L-top ological space and

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