In this paper we define concepts of statistical convergence and statistical Cauchy on probabilistic normed spaces. Then we give a useful characterization for statistically convergent sequences. Furthermore, we display an example such that our method of convergence is stronger than the usual convergence on probabilistic normed spaces. We also introduce statistical limit points, statistical clust...

It is shown that every fuzzy n-normed space naturally induces a locally convex topology, and that every finite dimensional fuzzy n-normed space is complete.

In this paper, we define multi-normed spaces, and investigate some properties of multi-bounded mappings on multi-normed spaces. Moreover, we prove a generalized Hyers– Ulam–Rassias stability theorem associated to the Cauchy additive equation for mappings from linear spaces into multi-normed spaces.

We show that if the Banach-Mazur distance between an n-dimensional normed space X and l∞ is at most 3/2, then there exist n+ 1 equidistant points in X. By a well-known result of Alon and Milman, this implies that an arbitrary n-dimensional normed space admits at least e √ log n equidistant points, where c > 0 is an absolute constant. We also show that there exist n equidistant points in spaces ...

In this article, we deal with weak convergence on sequences in real normed spaces, and weak* convergence on sequences in dual spaces of real normed spaces. In the first section, we proved some topological properties of dual spaces of real normed spaces. We used these theorems for proofs of Section 3. In Section 2, we defined weak convergence and weak* convergence, and proved some properties. By...

In this paper, we obtain a characterization of linear spaces which may be normed so as to become complete, linear, normed metric spaces. In this connection, K. Kunugui and M. Fréchet have shown that every metric space S is isometric with a subset of a complete, linear, normed metric space. I t follows from our result that if the cardinal number of 5 is the limit of a denumerable sequence of car...

In this paper, we introduce the cone normed spaces and cone bounded linear mappings. Among other things, we prove the Baire category theorem and the Banach--Steinhaus theorem in cone normed spaces.

In this paper, we define the notion of (dual) multi-fuzzy normedspaces and describe some properties of them. We then investigate Ulam-Hyers stability of Jensen's functional equation for mappings from linear spaces into  multi-fuzzy normed spaces. We establish an asymptotic behavior of the Jensen equation in the framework of multi-fuzzy normed spaces.