Narrow Results

This paper is about the generalization of the concept of fuzzy metric. Here we introduce g-fuzzy metric space of order $n$. In particular case, it reduces to George- and Veeramani-type fuzzy metric and Q-fuzzy metric. Some non-trivial examples of g-fuzzy metric space are given and its topological structure, the notion of convergent sequence, Cauchy sequence, etc. are also studied.

In our previous paper, it is shown that topology of G-fuzzy normed linear space is generated by two types of open balls: one is elliptic and the other is circular. In the theoretical aspect of functional analysis, will this type of exception happen or not? To address this problem in this paper, firstly, G-fuzzy bounded linear operators as well as G-fuzzy bounded linear functionals are defined which are the key elements of functional analysis. Then, operator G-fuzzy norms are introduced for both the cases using the idea of quasi-G-norm family. The definition of operator G-fuzzy norm is quite different from the existing operator fuzzy norm. Completeness of operator G-fuzzy norm is investigated. Lastly, Hahn-Banach theorem in G-fuzzy setting is studied using all the above concepts.

Following the concept of l-fuzzy convergent sequence, l-fuzzy closed, l-fuzzy complete, l-fuzzy compact sets in fuzzy normed linear spaces, in this paper, we study some results in finite-dimensional fuzzy strong $\varphi$-b-normed linear spaces and extend the most momentous Riesz Lemma.

In this paper, concept of fuzzy continuous operator, fuzzy bounded linear operator are introduced in fuzzy strong $\varphi$-b-normed linear spaces and their relations are studied. Idea of operator fuzzy norm is developed and completeness of BF(X,Y) is established.

This paper consists of the proof of the equivalence between two fuzzy strong $\varphi$-b-norms on finite-dimensional linear space and establish a Banach-type contraction theorem in this new setting.