局部凸L-拓扑向量空间的新定义与广义L-模糊半范数

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本文由张化朋和方锦暄两位学者合作撰写，旨在探讨局部凸L-拓扑向量空间的新定义。L-拓扑是模糊拓扑学在向量空间理论中的一个重要应用，它结合了拓扑结构与模糊集合的概念，使得在处理不确定性数据时具有更强的灵活性。在原有的研究中，严从华和方锦暄在其论文《L-fuzzy locally convex topological vector spaces》（J. Fuzzy Math., 7(3), 1999, 765-772）中提出了局部凸L-拓扑向量空间的定义，这为模糊分析领域提供了基础。本文首先给出了局部凸L-拓扑向量空间的新定义，这可能涉及对原有定义的改进或扩展，旨在更好地反映模糊环境下的局部凸性，即向量空间在每个点附近可以由一组L-fuzzy半范数（一种模糊版本的半范数，能够量化不确定度）来刻画其局部结构。新定义可能强调了模糊性与局部性质之间的更紧密联系，有助于处理模糊向量空间中的收敛性和稠密性问题。接着，作者引入了广义L-fuzzy半范数的概念，这是一种更为广泛的模糊半范数形式，可能允许更多的模糊性表达，从而增强了对局部凸性概念的理解和刻画。这一概念的引入对于理解模糊向量空间的性质至关重要，因为它提供了一种量化模糊元素之间关系的方式。利用这一系列的广义L-fuzzy半范数，作者给出了新定义下局部凸L-拓扑向量空间的特征描述，这可能是通过一系列模糊半范数的组合或者满足特定的模糊一致条件来实现的。这些描述为判断一个模糊向量空间是否具有局部凸性提供了清晰的判据。文章的最后部分，作者将这一新定义的应用深入到实际问题中，例如Hausdorff分离性质的研究。Hausdorff分离性质是拓扑空间的重要特性，模糊环境下的Hausdorff分离表明了空间中任意两点的模糊开集相互独立，这对于模糊数据处理中的分离和分类任务至关重要。此外，论文还探讨了分子网的收敛性以及L-fuzzy集在局部凸L-拓扑向量空间中的有界性问题，这些都是模糊数学中基础且实用的课题。总结来说，这篇文章不仅对局部凸L-拓扑向量空间进行了新的定义，而且还通过引入广义L-fuzzy半范数，对其性质进行了深入探讨，并将其应用于关键的数学概念如分离性和收敛性上，进一步丰富了模糊分析领域的理论基础和实际应用。

A is called semi-balanced if sA 6 A for all s ∈ K with 0 < |s| 6 1. If A

is semi-

balanced, we call A co-semi-balanced.

P ∈ L

is said to be co-normal if P (θ) 6 P (x) for all x ∈ X.

Deﬁnition 2.7 (Yan and Fang [10] ). Let (L

, δ) be an L-tvs. A family B of L-fuzzy

sets on X is called a prebase of δ if for each λ ∈ M(L), B

= {B

∨ µ

| B ∈ B, λ 66 µ}

is an R-neighborhood base of θ

. An L-tvs (L

, δ) is said to be locally convex, if there

exists a family B consisting of convex and balanced L-fuzzy sets on X such that B is a

prebase of δ.

In the case of L = [0, 1], locally convex L-tvs is exactly the locally convex I-tvs deﬁned

by Wu and Li [8].

Lemma 2.1 (Fang and Yan [2] ). Let (L

, δ) be an L-tvs, and B

(λ ∈ M(L)) be a

closed R-neighborhood base of θ

. Then B

(λ ∈ M(L)) has the following properties:

(1) If W ∈ B

or W = λ

, then for any x ∈ X and α ∈ M(L) with x

 W , there

exists P ∈ B

such that W 6 x + P ;

(2) If P, Q ∈ B

, then there exists W ∈ B

such that P ∨ Q 6 W ;

(3) If W ∈ B

, then there exists P ∈ B

such that P

+ P

6 W

;

(4) If W ∈ B

, then there exists P ∈ B

such that tP

6 W

for all t ∈ K with

|t| 6 1;

(5) If W ∈ B

, then for each x ∈ X there exists t > 0 such that x

 tW .

Conversely, if for each λ ∈ M(L) there is a family B

of L-fuzzy sets on X satisfying the

above conditions (1)−(5), then there exists a unique L-topology δ on X such that (L

, δ)

is an L-tvs, and B

is a closed R-neighborhood base of θ

Lemma 2.2 (Fang and Yan [2] ). Let (L

, δ) be an L-tvs. If Q ∈ δ, then x + Q ∈ δ for

each x ∈ X and kQ ∈ δ for all k ∈ K \ {0}.

Lemma 2.3 (Fang and Yan [2] ). Each L-tvs (L

, δ) has an R-neighborhood base of θ

consisting of co-balanced L-fuzzy sets on X for each λ ∈ M (L).

Lemma 2.4 (Yan and Fang [11] ). If A ∈ L

is balanced, then conv(A) is also balanced.

Lemma 2.5. Let (L

, δ) be an L-tvs, P ∈ L

and Q ∈ δ. Then P + Q ∈ δ.

Proof. If P = 0

, then P + Q = 0 ∈ δ; If P 6= 0, we let S

= {x ∈ X | P (x) 6= 0}, then

it is easy to see that P =

x∈S

P (x)

. Hence

P + Q =



x∈S

P (x)



+ Q =

x∈S



P (x)

+ Q



x∈S



P (x)

∧ (x + Q)



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