Quasilinear Inner Product Spaces and Hilbert Quasilinear Spaces

Abstract

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 Başar (2010), and Nikol'skiȋ (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.

New Inner Product Quasilinear Spaces on Interval Numbers

“…Then we know that Ω( ) is an IPQLS and it is complete with respect to the Hausdorff metric. So, Ω( ) is a Hilbert QLS [4].…”

“…We also say that and are orthogonal and we write ⊥ . Similarly, for subsets , ⊆ we write ⊥ if ⊥ for all ∈ and ⊥ if ⊥ for all ∈ and ∈ [4].…”

“…Inspired and motivated by research going on in this area, we introduce an inner product quasilinear space which is defined in [4]. Generally, in [4], we give some examples of quasi-inner product properties on Ω ( ) of all convex compact subsets of a normed space . In our present paper we examine a new type of a quasilinear space, namely, R .…”

On the Algebra of Interval Vectors

“…In our [9][10][11][12] referenced articles, we tried to eliminate some of these shortcomings in quasilinear algebra. Then we also introduced the concept of inner product in quasilinear spaces, and thus we were able to define the concept of Hilbert quasilinear space definition [13][14][15][16]. The introduction of these concepts also provides us with the opportunity to make many applications.…”

Normed proper quasilinear spaces

“…Thus his treatment allows us to construct a kind of theory of quasilinear algebra. Aseev's avant garde work has motivated us to introduce some new results, [1,3,4,5,6,7,8,9,10,11].…”