Some properties of orthonormal sets on inner product quasilinear spaces

Aseev introduced the notion of quasilinear space and normed quasilinear space [1]. Then he gives some basic definition and theorems related to these spaces. We can see in [1] that, as different from linear spaces, Aseev used the partial order relation when he defined quasilinear spaces and so he can give consistent counterparts of results in linear spaces. This pioneering work has motivated us to introduce the concept of inner product quasilinear spaces and Hilbert Quasilinear spaces which were given in [5]. Our research with these spaces continued to investigate the quasilinear counterparts of fundamental theorems in linear functional analysis. While working in this nonlinear spaces, we have noticed that the new definition is necessary. So, in this study, we give some new concepts related to inner product quasilinear spaces. In this paper, we present some properties of orthogonal and orthonormal sets on inner product quasilinear spaces. We introduce solid floored quasilinear spaces to deal with a better and suitable kind of quasilinear spaces. To define some topics such as Schauder basis, complete orthonormal sequence, orthonormal basis and complete set and some related theorems this classification is crucial. Furthermore, we try to display some geometric differences of inner product quasilinear spaces from the inner product (linear) spaces.