Generalized Fuzzy Normed Normed Space And Operator Theory

In this article, using the idea of fuzzy sets, we define the notion of generalized fuzzy normed spaces in light of continuous t-norm. Some properties of generalized fuzzy normed spaces are discussed. There are four chapters as follows.In Chapter 1, we first recall the concepts of fuzzy norm, continuous t-norm and fuzzy normed space. Then the definitions of generalized fuzzy norm and generalized fuzzy normed space are introduced. Also, continuity of the generalized fuzzy norm is discussed. Finally we define the notions of fuzzy convergence, Cauchy sequence and fuzzy bounded set and prove several properties of them.In Chapter 2, the definitions of fuzzy open ball and fuzzy closed ball are first introduced and some properties of them are proved. After that, fuzzy open and fuzzy closed sets are defined which can induce a topology on a generalized fuzzy normed space. Finally two conclusions about complete spaces are proved by the concept of nest of fuzzy closed balls in a generalized fuzzy normed space.In Chapter 3, the notions of relatively sequentially fuzzy compact and sequentially fuzzy compact sets are first given and the relation between fuzzy closedness and fuzzy boundedness is considered. Then we define fuzzy compact set, (ε,t)-nest and fuzzy totally bounded set by the idea of cover. We also explore the relation between fuzzy totally boundedness and fuzzy boundedness and get an equivalent condition of fuzzy totally boundedness. At last, we prove that fuzzy compactness and sequentially fuzzy compactness are equivalent and fuzzy totally bounded sets are seperable.In Chapter 4, we define the notions of fuzzy continuous operator and sequentially fuzzy continuous operator in Section 1 and get two equivalent conditions first. Then the following conclusions are proved:the composition of fuzzy continuous operators is fuzzy continuous, fuzzy continuous operator is equivalent to sequentially fuzzy continuous operator, the operator which is sequentially fuzzy continuous operator at a point is sequentially fuzzy continuous operator on the whole space and the image of a fuzzy compact set under a fuzzy continuous operator is fuzzy compact. In Section 2, the notion of fuzzy compact operator is given and we get an equivalent condition of fuzzy compact operator. Finally we prove the composition of fuzzy compact operator and fuzzy continuous operator is fuzzy compact.