In this paper, the gradual real numbers are considered and the notion of the gradual normed linear space is given. Also some topological properties of such spaces are studied, and it is shown that the gradual normed linear space is a locally convex space, in classical sense. So the results in locally convex spaces can be translated in gradual normed linear spaces. Finally, we give an example of a gradual normed linear space which is not normable in classical analysis.

In this paper, we choose an approach to generalize the idea  of usual norm keeping up  analogous structure and introduce a  generalized norm, called  $ \Phi $-norm using a  function $ \Phi $ which satisfies some special condition. We show that the induced metric of it is $ \phi $-metric. We also study some basic properties and obtain some fundamental results in such  spaces.

In this paper, we shall define and study the concept of l-statistical convergence and l-statistical Cauchy in random 2-normed space. We also introduce the concept of l-statistical completeness which would provide a more general frame work to study the completeness in random 2-normed space. Furthermore, we also prove some new results.

Recently, Mursaleen introduced the concepts of statistical convergence in random 2-normed spaces. In this paper, we define and study the notion of L-statistical convergence and L-statistical Cauchy sequences in random 2-normed spaces , where  l=(lm) be a non-decreasing sequence of positive numbers tending to infinity such thatlm+1£lm + 1, l1=1 and prove some theorems. In last section we will give the definition of the L-limit and cluster points and we will show their relation between those classes.

Let X be a real 2-Banach space. We follow Gunawan, Mashadi, Gemawati, Nursupiamin and Siwaningrum in saying that x is orthogonal to y if there exists a subspace V of X with codim(V ) = 1 such that ∥, x+λ, y,z∥,≥,∥, x,z∥,for every z ∈,V and λ,∈,R. In this paper we prove that every linear self mapping T: X →,X which preserve orthogonality is a 2-isometry multiplied by a constant.

Following the definition of interval-valued fuzzy n-normed linear space given by S. Vijayabalaji et al., in this paper, the notion of interval-valued generalized fuzzy n-normed linear space is introduced. The notion of convergent sequence, Cauchy sequence and their relation are studied. Some basic results are established on finite-dimensional interval-valued generalized fuzzy n-normed linear space.

In this paper, we define strong deferred summability and deferred statistical summability in intuitionistic fuzzy \(n\)-normed linear spaces (briefly called IF-\(n\)-NLS) and study some of their properties. We also define deferred statistical Cauchy sequence deferred statistical completeness and characterize deferred statistical summability in such spaces. Finally, we will compare deferred statistical summability for different pairs of sequences $\alpha(\ell)$, $\gamma(\ell)$,$u(\ell)$ and $v(\ell)$ satisfying $\alpha(\ell) \leq u(\ell) < v(\ell) \leq \gamma(\ell)$.