Some Studies On Linear Operators In Fuzzy Quasi-Normed Spaces

In 2004,Gregori and Romaguera proposed the concept of fuzzy quasi-norm by combining fuzzy set and quasi-norm.Then the research of fuzzy quasi-norm space theory begins.After more than 10 years of exploration,the research in this field has made some progress,and it shows important application prospect in many aspects including the research of algorithm complexity.Linear operator theory is an important part of linear functional analysis.Alegre and Romaguera have extended the uniformly bounded principle of linear functional analysis to fuzzy quasi-normed spaces.Inspired by this,this paper makes further research on linear operator theory in fuzzy quasi-normed space.The main contents are included:First,we first study continuous linear functionals on a fuzzy quasi-normed space,obtain a characterization of continuous linear functionals,and point out that the set of all continuous linear functionals forms a convex cone and can be equipped with a weak fuzzy quasi-norm.Next,we prove a theorem of Hahn-Banach type and two separation theorems for convex subsets of fuzzy quasi-normed spaces.Secondly,we study some topological properties of fuzzy quasi-normed spaces,and prove the open mapping and the closed graph theorems in the framework of fuzzy quasinormed spaces.The results obtained in this paper enrich the theory of fuzzy quasi-normed space,especially the study of linear operators in fuzzy quasi-normed space.It is of great significance to promote the theoretical development and application of fuzzy analysis.