J. Appl. Math. & Computing Vol. 17(2005), No. 1 - 2, pp. 475 - 484 SOME RESULTS ON FUZZY BANACH SPACES R. SAADATI AND S. M. VAEZPOUR * Abstract. The main aim of this paper is to consider the fuzzy norm, define the fuzzy Banach spaces, its quotients and prove some theoremes and in particular Open mapping and Closed graph theoremes on these spaces. AMS Mathematics Subject Classification : 46S40. Key words and phrases : Fuzzy space, fuzzy normed space, fuzzy Banach space. 1. Introduction The theory of fuzzy sets was introduced by L. Zadeh [8] in 1965. Many mathematicians considered the fuzzy metric spaces in different view ( see [1], [3], [4] and [5]). First we recall the definition of continuous t-norm, fuzzy metric spaces and Cauchy sequences introduced by George and Veermani [3]. Definition 1. A binary operation ∗ : [0, 1] × [0, 1] -→ [0, 1] is said to be a continuous t-norm if ([0, 1], ∗) is a topological monoid with unit 1 such that a ∗ b ≤ c ∗ d whenever a ≤ c and b ≤ d (a, b, c, d ∈ [0, 1]). It is clear that if we define a ∗ b = ab or a ∗ b = min(a, b) then ∗ is a continuous t-norm. Definition 2. The 3-tuple (X,M, ∗) is said to be a fuzzy metric space if X is an arbitrary set, ∗ is a continuous t-norm and M is a fuzzy set on X 2 × (0, ∞) satisfying the following conditions for all x, y, z ∈ X and t, s > 0, (i) M (x, y, 0) > 0, Received October 21, 2003. Revised April 11, 2004. * Corresponding author. c  2005 Korean Society for Computational & Applied Mathematics and Korean SIGCAM. 475