On orthonormal sets in inner product quasilinear spaces

Aseev, S. M [Aseev, S. M., Quasilinear operators and their application in the theory of multivalued mappings, Proc. Steklov Inst. Math., 2 (1986), 23–52] generalized linear spaces by introducing the notion of quasilinear spaces in 1986. Then, special quasilinear spaces which are called “solid floored quasilinear spaces” were defined and their some properties examined in [Çakan, S., Some New Results Related to Theory of Normed Quasilinear Spaces, Ph.D. Thesis, Inönü University,
Malatya, 2016]. In fact, this classification was made so as to examine consistent and detailed some properties related to quasilinear spaces.
In this paper, we present some properties of orthogonal and orthonormal sets on inner product quasilinear spaces. At the same time, the mentioned classification is crucial for define some topics such as Schauder basis, complete orthonormal sequence, orthonormal basis and complete set and some related theorems. Also, we try to explain some geometric differences of inner product quasilinear spaces from the inner product (linear) spaces.