Some Properties Of Cartesian Product Of Two Fuzzy Normed Spaces

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.