Some new results related to the theory of soft normed quasilinear spaces

This master’s thesis, named “Some New Results Related to the Theory of Soft Normed Quasilinear Spaces”, mainly consists of five chapters. In the first chapter, information about the content of the thesis is given. In the second part, some basic concepts, theorems and results of linear functional analysis, which will be used in the following parts of this study, and Hausdorff distinction, distance and metric are outlined. The third chapter of the thesis consists of basic definitions and results in quasilinear and normed quasilinear spaces. In the fourth chapter, first of all, soft sets and some of their properties are introduced, then some basic concepts and theorems about soft linear spaces and soft quasilnear spaces to be used in the fifth part of the thesis are given. The fifth chapter, which is the original part of the thesis, is divided into four subsections. In the first subsection, some results about the convergence of a soft quasi sequence defined in a quasilinear space with a soft norm are given and the completeness of a quasilinear space with a soft norm is examined. In the second subsection, the concepts of soft quasilinear independence and dependence in soft normed quasilinear spaces, just like in normed quasilinear spaces, are defined and their consistent equivalents in quasilinear spaces are obtained. In the third subsection, soft proper quasilinear spaces, which are a generalization of proper quasilinear spaces, are examined. In the fourth subsection, a quasilinear functional with a soft norm is defined in a quasilinear space with a soft norm and some results are obtained about this new functional. Some theorems about the continuity and limitation of soft quasilinear operators and functionals have been proved.