Quasilinear Inner Product Spaces and Hilbert Quasilinear Spaces

Aseev launched a new branch of functional analysis by introducing the theory of quasilinear spaces in the framework of the topics of norm, bounded quasilinear operators and functionals (Aseev (1986)). Furthermore, some quasilinear counterparts of classical nonlinear analysis that lead to such result as Frechet derivative and its applications were examined deal with. This pioneering work causes a lot of results in such applications such as (Rojas-Medar et al. (2005), Talo and Ba？ar (2010), and Nikol'ski？ (1993)). His work has motivated us to introduce the concept of quasilinear inner product spaces. Thanks to this new notion, we obtain some new theorems and definitions which are quasilinear counterparts of fundamental definitions and theorems in linear functional analysis. We claim that some new results related to this concept provide an important contribution to the improvement of quasilinear functional analysis. 1. Introduction The theory of quasilinear space was introduced by Aseev [1]. He proceeds, in a similar way to linear functional analysis on quasilinear spaces by introducing the notions of norm, with quasilinear operators and functionals. 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. Further we note that the norm defined in quasilinear space is compatible with the concept of norm on linear space and if each element of normed quasilinear space has an inverse, then the partial order is determined by equality. Consequently, the concept of normed quasilinear spaces coincides with the concept of normed space in classical analysis. As known, the theory of inner product space and Hilbert spaces play a fundamental role in functional analysis and its applications such as integral and differential equations, approximation theory, linear and nonlinear stability problems, and bifurcation theory. We know that any inner product space is a normed space and any normed space is a particular class of normed quasilinear space. Hence, this relation and Aseev's work motivated us to examine quasilinear counterparts of inner product space in classical analysis. Thus, we introduce the concept of quasilinear inner product space as a new structure. Moreover, we obtain some definitions and results related to this notion which provide us with improving the elements of the quasilinear functional analysis. The definition of quasi-inner product function is extended by classical definition of inner product function. It is