Abstract
In the present paper some different types of boundedness in fuzzy normed linear spacesof type (X, N, ∗), where is an arbitrary t-norm, are considered. These boundedness concepts arevery general and some of them have no correspondent in the classical topological metrizable linearspaces. Properties of such bounded sets are given and we make a comparative study among thesetypes of boundedness. Among them there are various concepts concerning symmetrical properties ofthe studied objects arisen from the classical setting appropriate for this journal topics. We establishthe implications between them and illustrate by examples that these concepts are not similar.
Highlights
Briefly FNL spaces, were first introduced by Katsaras, who introduced some general types of fuzzy topological linear spaces [1,2]
Introduced another concept of fuzzy norm defined on a vector space by putting in correspondence to each element of the linear space, a fuzzy real number
As any fuzzy norm induces naturally a fuzzy metric, for studying boundedness we can use the notion of F-bounded introduced by George and Veeramani for fuzzy metric spaces
Summary
Briefly FNL spaces, were first introduced by Katsaras, who introduced some general types of fuzzy topological linear spaces [1,2]. Inspired by Cheng and Mordeson [4], in 2003, Bag and Samanta [5] defined a more suitable notion of fuzzy norm, even if it could be more refined, made simpler or even made more general (see [6,7,8,9,10]) In this context, there are two concepts of boundedness, one of them introduced by Bag and Samanta [5] and the other one introduced by Sadeqi and Kia [11] in 2009. The paper comprises the following: we begin with the preliminary section, in Section 2, we study fuzzy bounded sets This concept of boundedness corresponds to the classical boundedness, as it is shown in Theorem 4. In Proposition 12 is given an example of a fuzzy bounded set that is not fuzzy totally bounded
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