Equivalent normed quasilinear spaces

Aseev introduced the concepts of quasilinear spaces and normed quasilinear spaces, in his article [1]. He used the partial order relation to define quasilinear spaces. He stated properties and results which are quasilinear counterparts of some results in classical linear functional analysis. Also, in [1], he defined the some new concepts which are only meaningful in normed quasilinear spaces. We give, in this presentation, some new results and examples on quasilinear spaces and normed quasilinear spaces. Further, we introduce the concept of equivalent norms on a quasilinear space. By novelty of the new definition, we state on the Hausdorff metric properties for equivalent norms which are extend to the quasilinear context some results of linear functional analysis.