Some Properties Of Cartesian Product Of Two Fuzzy Normed Spaces

Abstract

In this paper , the concept of the Cartesian Product of two fuzzy
 normed spaces is presented. Some basic properties and theorems on this
 concept are proved. The main goal of this paper is to prove that the
 Cartesian product of two complete fuzzy normed spaces is a complete
 fuzzy normed space.
 Key words:Fuzzy normed space , Cartesian product , Cauchy sequence ,
 complete fuzzy normed space.
 1- Introduction
 
 The fuzzy set concepts was introduced in mathematics by K.Menger
 in 1942 and reintroduced in the system theory by L.A.Zadeh in 1965.
 In 1984, Katsaras [ 1 ] , first introduced the notation of fuzzy norm on
 linear space, in the same year Wu and Fang [ 4 ] also introduced a notion of
 fuzzy normed space . Later on many other mathematicians like Felbin [ 2 ]
 , Cheng and Mordeson [ 10] , Bag and Samanta [12], J.Xiao and X.Zhu
 [8,9] , Krishna and Sarma [11] , Balopoulos and Papadopoulos [ 13] etc,
 have given different definitions of fuzzy normed spaces .
 J.Kider introduced the definition of fuzzy normed space[ 7 ] , we use this
 definition to prove that the Cartesian product of two fuzzy normed spaces
 is also fuzzy normed space.
 
 The structure of the paper is as follow : In section 2 we
 present some fundamental concepts . In section 3, the definition of fuzzy
 normed space appeared [7] is used to prove that the cartesain product of
 two fuzzy normed spaces is also fuzzy normed space, then we prove that
 the cartesain product of two complete fuzzy normed spaces is complete
 fuzzy normed space.
 
 2. Preliminaries
 In this section, we briefly recall some definitions and preliminary
 results which are used in this paper.