This paper generalizes the Aleksandrov problem, the Mazur–Ulam theorem and Benz theorem on n-normed spaces. It proves that a one-distance preserving mapping is an nisometry if and only if it has the zero-distance preserving property, and two kinds of n-isometries on n-normed spaces are equivalent.

In this paper we introduce the sequence spaces χM (p, q, u), using an modulus function M and defined over a semi normed space (X, q), semi normed by q. We study some properties of these sequence spaces and obtain some inclusion relations.

is finite, where λ denotes Lebesgue measure. Let ρ be the gauge functional of the convex hull of the unit ball {f : q(f) ≤ 1} of the quasi-norm q, and let N be the null space of ρ. The normed envelope of WeakL, which we denote by W , is the space (WeakL/N, ρ). The Banach envelope of WeakL, W , is the completion of W . We show that W is isometrically lattice isomorphic to a sublattice of W . It ...

Recently in [22], Mursaleen introduced the concept of statistical convergence in random 2-normed spaces. In this paper, we define and study the notion of lacunary ∆-statistical convergence and lacunary ∆-statistical Cauchy sequences in random 2-normed spaces using lacunary density and prove some interesting theorems. Subjclass [2000] : Primary 40A05; Secondary 46A70, 40A99, 46A99.

Copyright q 2010 Dongseung Kang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We investigate the generalized Hyers-Ulam-Rassias stability problem in quasi-β-normed spaces and then the stability by using a subadditive function f...

K?nzi and Yilzid introduced the concept of convexity structures in sense Takahashi quasi-pseudometric spaces [7]. In this article, we continue study theory, introducing W-convexity for real-valued pair functions defined on an asymmetrically normed real vector space. Moreover, show that all minimal pairs space equipped with a convex structure which is W-convex whenever W translation-invariant.

in this paper, we obtain the general solution and the generalized  hyers--ulam--rassias stability in random normed spaces, in non-archimedean spacesand also in $p$-banach spaces and finally the stability viafixed point method for a functional equationbegin{align*}&d_f(x_{1},.., x_{m}):= sum^{m}_{k=2}(sum^{k}_{i_{1}=2}sum^{k+1}_{i_{2}=i_{1}+1}... sum^{m}_{i_{m-k+1}=i_{m-k}+1}) f(sum^{m}_{i=1...

w0-Nearest Points and w0-Farthest Point in Normed Linear Spaces

We study computability on sequence spaces, as they are used in functional analysis. It is known that non-separable normed spaces cannot be admissibly represented on Turing machines. We prove that under the Axiom of Choice non-separable normed spaces cannot even be admissibly represented with respect to any compatible topology (a compatible topology is one which makes all bounded linear function...