Geometric properties of a novel type of orthogonality via norm derivatives

Abstract

In this article, we generalize the notion of orthogonality as a linear combination of norm derivatives in order to give a novel concept that we refer to as ${\rho }_{\alpha ,\beta }$-orthogonality. Also, we discuss some of its geometric properties in a real normed linear space and present some sufficient criteria for the smoothness of a normed space by using ${\rho }_{\alpha ,\beta }$-orthogonality. We provide a few examples to show that the ${\rho }_{\alpha ,\beta }$- orthogonality cannot be compared to other well-known orthogonalities in any way. In addition to this, we offer a characterization of inner product spaces by making use of the functional notation ${\rho }_{\alpha ,\beta }$. In addition, we show that any ${\rho }_{\alpha ,\beta }$-orthogonality that preserves linear mapping between two normed linear spaces must necessarily be a scalar multiple of an isometry. Also, using the ${\rho }_{\alpha ,\beta }$-functional, we define the idea of an angle between two vectors and talk about their characteristics in normed spaces.