In this article, we considered bidual spaces and reflexivity of real normed spaces. At first we proved some corollaries applying Hahn-Banach theorem and showed related theorems. In the second section, we proved the norm of dual spaces and defined the natural mapping, from real normed spaces to bidual spaces. We also proved some properties of this mapping. Next, we defined real normed space of R...

In this paper, we have generalized the definition of vector space by considering the group as a canonical $m$-ary hypergroup, the field as a krasner $(m,n)$-hyperfield and considering the multiplication structure of a vector by a scalar as hyperstructure. Also we will be consider a normed $m$-ary hypervector space and introduce the concept of convergence of sequence on $m$-ary hypernormed space...

Given a normed cone (X , p) and a subcone Y, we construct and study the quotient normed cone (X/Y, p̃) generated by Y . In particular we characterize the bicompleteness of (X/Y, p̃) in terms of the bicompleteness of (X , p), and prove that the dual quotient cone ((X/Y )∗,‖ · ‖p̃,u) can be identified as a distinguished subcone of the dual cone (X∗,‖ · ‖p,u). Furthermore, some parts of the theory ar...

Summary This paper formalizes in Mizar [1], [2], that the isometric isomorphisms between spaces formed by an ( n + 1)-dimensional multilinear map and -fold composition of linear maps on real normed spaces. result is used to describe space nth-order derivatives Frechet derivative as a space. In Section 1, we discuss 1-dimensional 0-fold compositions preparation, 2, extend discussion compositions...

We study new sequence spaces associated to sequences in normed spaces and the band matrix F̂ defined by the Fibonacci sequence. We give some characterizations of continuous linear operators and weakly unconditionally Cauchy series by means of completeness of the new sequence spaces. Also, we characterize the barreledness of a normed space via weakly∗ unconditionally Cauchy series in [Formula: se...

In this paper we define the concepts of λ-statistical convergence and λ-statistically Cauchy in probabilistic normed space and prove some interesting results. Furthermore, we display an example such that our method of convergence is stronger than the usual convergence in probabilistic normed spaces.

In this paper, using the direct method we study the generalized Hyers-Ulam-Rassias stability of the following quadratic functional equations (2 ) ( ) 6 ( )     f x y f x y f x and (3 ) ( ) 16 ( )     f x y f x y f x for the mapping f from normed linear space in to 2-Banach spaces.

In normed linear space settings, modifying the sequential definition of continuity of an operator by replacing the usual limit "lim" with arbitrary linear regular summability methods G we consider the notion of a generalized continuity ((G1,G2)-continuity) and examine some of its consequences in respect of usual continuity and linearity of the operators between two normed linear spaces.

The main result of this article is Theorem 1 which states that a quotient space Y, dim Y = k, of a subspace of any finite dimensional normed space X, dim X — n, may be chosen to be J-isomorphic to a euclidean space even for k = [Xn] for any fixed X < 1 (and d depending on X only). The following theorem is proved. 1. Theorem. For every d > 1 there exists X(d) > 0 such that every n-dimensional no...