Finite Dimensional Generating Spaces of Quasi-Norm Family

Abstract

In this paper, some results on nite dimensional generating spaces of quasi-norm family are established. The idea of equivalent quasi-norm fami- lies is introduced. Riesz lemma is established in this space. Finally, we re-dene B-S fuzzy norm and prove that it induces a generating space of quasi-norm family. Chang et al.(2) dealt with the generating spaces of quasi-metric family and by studying the common characteristic properties of both fuzzy metric spaces in the sense of Kaleva & Seikkala (6) and Menger probabilistic metric spaces(11) estab- lished that the generating spaces of quasi-metric family possess the unification characterization of the fuzzy and probabilistic situations. They also proved sev- eral fixed point theorems in generating spaces of quasi-metric family. Jung et al.(5) established some fixed point theorems in generating spaces of quasi-metric family. Many authors introduced different ideas of fuzzy norm ((9), (10), (13)), fuzzy n-norm (3), fuzzy Hilbert space(4) etc. in different ways. In 2006, Xiao & Zhu(12) introduced a concept of a generating space of quasi-norm family (G.S.Q.N.F) and studied its properties in a linear topological space setting. They also introduced the concept of convergent sequences, Cauchyness, completeness, compactness, etc. and extended some fixed point theorems, especially Schauder-type fixed point theorem in such spaces. In this paper, we study certain results in finite dimensional generating spaces of quasi-norm family (G.S.Q.N.Fs). We observe that most of these results more or less differ from the results in finite dimensional normed linear spaces. We generalize the definition of fuzzy norm as introduced by Bag & Samanta(1) (B-S fuzzy norm), and we deduce a G.S.Q.N.F from it. The organization of the paper is as follows: Section 1 comprises of some preliminary results. In section 2, we study some results in finite dimensional G.S.Q.N.Fs. In section 3, a generalized form of B-S fuzzy norm is advanced and a G.S.Q.N.F is deduced from it. Throughout this paper straightforward proofs are omitted.