~fuzzy norm

‎‎‎Given a fuzzy normed space, we will introduce the notion of fuzzy near best approximation as a generalization of the notion of fuzzy best approximation. Some basic properties are characterized and also many examples for illustration are presented. Also, the hereditary properties of the fuzzy near best approximation on direct sum and tensor product of linear spaces are discussed.

For a nonzero normed linear space X, we will define some different classes of fuzzy norms on X generated by linear and bounded linear functionals. Also separate continuity of the elements within each class are investigated. The aim of this research is to introduce a source of examples and counterexamples in the field of fuzzy normed spaces.

In this paper, the Hyers--Ulam--Rassias stability of Jensen's functional equation is discussed in detail on fuzzy normed linear spaces. It is shown through the analysis that every fuzzy approximate Jensen-type mapping can be approximated by an additive mapping. The obtained results are the fuzzy versions of some classical theorems.

Following the definition of fuzzy normed linear space which was introduced by Bag and Samanta in general t-norm settings, in this paper,  definition of fuzzy strong $\phi$-b-normed linear space is given. Here the scalar function $|c| $ is replaced by a general function $ \phi(c) $ where $ \phi $ satisfies some properties. Some basic results on finite dimensional  fuzzy strong $\phi$-b-normed linear space are studied.

Three types of fuzzy topologies defined on fuzzy normed linear spaces are considered in this paper. First, the relationshipbetween fuzzy continuity of linear operators and fuzzy boundedness is investigated. The uniform boundedness theorem is then discussed, so too is the norm of a linear operator. Finally, the open mapping theorem is proved for each of the three defined fuzzy topologies, and the closed graph theorem is studied.

In the current paper, consider the fuzzy normed linear space $(X,N)$ which is defined by Bag and Samanta. First, we construct a new fuzzy topology on this space and show that these spaces are Hausdorff locally convex fuzzy topological vector space. Some necessary and sufficient conditions are established to illustrate that the presented fuzzy topology is equivalent to two previously studied fuzzy topologies.

ýIn this paperý, ýwe use the definition of fuzzy normed spaces given by Bag and Samanta and provide four types of fuzzy versions of contractioný. ýWe show that these mappings necessarily have unique fixed points in fuzzy normed linear spacesý. ýWe will show that the presented theorems are indeed fuzzy extensions of their classical counterpartsý.

In this paper, the connection between Menger probabilistic norms and H"{o}hle probabilistic norms is discussed. In addition, the correspondence between probabilistic norms and Wu-Fang fuzzy (semi-) norms is established. It is shown that a probabilistic norm (with triangular norm $min$) can generate a Wu-Fang fuzzy semi-norm and conversely, a Wu-Fang fuzzy norm can generate a probabilistic norm.

In the present paper, we investigate a connection between two fuzzy inner product one of which arises from Felbin's fuzzy norm and the other is based on Bag and Samanta's fuzzy norm. Also we show that, considering a fuzzy inner product space, how one can construct another kind of fuzzy inner product on this space.

In the present paper, we rst modify the concepts of weakly fuzzy boundedness, strongly fuzzy boundedness, fuzzy continuity, strongly fuzzy continuity and weakly fuzzy continuity. Then, we try to nd some relations by making a comparative study of the fuzzy norms of linear operators.

In the present paper, we study some properties of fuzzy norm of linear operators. At first the bounded inverse theorem on fuzzy normed linear spaces is investigated. Then, we prove Hahn Banach theorem, uniform boundedness theorem and closed graph theorem on fuzzy normed linear spaces. Finally the set of all compact operators on these spaces is studied.

In this paper,~some results on finite dimensional generating spaces of quasi-norm family are established.~The idea of equivalent quasi-norm families is introduced.~Riesz lemma is established in this space.~Finally,~we re-define B-S fuzzy norm and prove that it induces a generating space of quasi-norm family.

In this paper, we use the de nition of fuzzy normed spaces given by Bag and Samanta and the behaviors of solutions of the additive functional equation are described. The Hyers-Ulam stability problem of this equation is discussed and theorems concerning the Hyers-Ulam-Rassias stability of the equation are proved on fuzzy normed linear space.

In this paper, a fuzzy version of the analytic form of Hahn-Banach extension theorem is given. As application, the Hahn-Banach theorem for $r$-fuzzy bounded linear functionals on $r$-fuzzy normed linear spaces is obtained.