The main purpose of this paper is to consider the t-best simultaneousapproximation in fuzzy normed spaces. We develop the theory of t-bestsimultaneous approximation in quotient spaces. Then, we discuss the relationshipin t-proximinality and t-Chebyshevity of a given space and its quotientspace.

In this paper we obtain some inequalities related to the generalized triangle and quadratic triangle inequalities for vectors in inner product spaces. Some results that employ the Ostrowski discrete inequality for vectors in normed linear spaces are also obtained.

Let X be a linear space. A p-norm on X is a real-valued function on X with 0 < p ≤ 1, satisfying the following conditions: (i) ‖x‖p ≥ 0 and ‖x‖p = 0⇔ x = 0, (ii) ‖αx‖p = |α|p‖x‖p, (iii) ‖x+ y‖p ≤ ‖x‖p +‖y‖p for all x, y ∈ X and all scalars α. The pair (X ,‖,‖p) is called a p-normed space. It is a metric linear space with a translation invariant metric dp defined by dp(x, y)= ‖x− y‖p for all x, ...

We prove a subgaussian extension of a Gaussian result on embedding subsets of a Euclidean space into normed spaces. Using the concentration of a random subgaussian vector around its mean we obtain an isomorphic (rather than almost isometric) result, under an additional cotype assumption on the normed space considered.

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.

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.

We investigate a notion of non-Archimedean fuzzy anti-n-normed spaces and prove the Mazur-Ulam theorem in the spaces. AMS subject classification: primary 46S10; secondary 47S10; 26E30; 12J25.

We show that quasihyperbolic geodesics exist in convex domains in reflexive Banach spaces and that quasihyperbolic geodesics are quasiconvex in the norm metric in convex domains in all normed spaces. 2000 Mathematics Subject Classification: 30C65