Fuzzy Normed Linear Spaces

Abstract

In this chapter we will present different concepts of fuzzy norms on a linear space, introduced by various authors from different points of view. Thus, in 1984, Katsaras was the first who introduced the notion of fuzzy norm, this being of Minkowsky type, associated to an absolutely convex and absorbing fuzzy set. In 1992, Felbin introduced another ideea of fuzzy norm, by assigning a fuzzy real number to each element of the linear space. Following Cheng and Mordeson, in 2003, Bag and Samanta proposed another concept of fuzzy norm, which will be proven most adequate, even if it can be still improved, simplified or generalized. In this context, this chapter will contain the results obtained by Nadaban and Dzitac in the paper “Atomic decomposition of fuzzy normed linear spaces for wavelet applications”. We also note that a new concept of fuzzy norm was introduced by Saadati and Vaezpour, in 2005. The concept of fuzzy norm has been generalized to continuous t-norm by Goleţ in 2010. Recently, Alegre and Romaguera proposed the term of fuzzy quasi-norm and Nadaban introduced the notion of fuzzy pseudo-norm.