Some new results related to theory of normed Quasilinear spaces

In the fi rst chapter of this thesis consisting of five chapters, a brief overview of the literature is presented. In the second chapter, some fundamental defi nitions, theorems and results used in the next chapters are given. In the third chapter, after introducing the concepts of regular and singular dimension of a quasilinear space and proper quasilinear space, some basic theorems and results about these concepts are obtained. Also, some evaluations about the theories of finite regular and singular dimensional normed quasilinear spaces and normed proper quasilinear spaces have been made. Next, special proper quasilinear spaces which are called solid fl oored quasilinear spaces are defi ned and their some properties are examined. In the fourth chapter, one of the general methods to construct a topology on any quasilinear space is presented and the necessary preparations for this method are stated. In this chapter, it is proved that the important property of the normed linear spaces namely the localization principle may not hold in normed quasilinear spaces. Then, it has been tried to draw a border of the localization principle in the normed quasilinear spaces. In the fifth chapter, the concepts of lower and upper semi convergence of a sequence in normed quasilinear spaces are introduced and some results related to these concepts are obtained. KEY WORDS: Quasilinear spaces, Normed quasilinear spaces, Hausdorff metric, Regular and singular dimension, Floor of an element, Proper quasilinear spaces, Solid floored quasilinear spaces, Stu ff of an element, Localization principle, Lower and upper semi convergence.