Geometric properties of a novel type of orthogonality via norm derivatives

TL;DR

This paper introduces a new type of orthogonality in normed spaces based on norm derivatives, explores its geometric properties, and characterizes inner product spaces using this concept.

Contribution

It proposes the $ ho_{eta,eta}$-orthogonality, analyzes its properties, and provides criteria for smoothness and inner product space characterization.

Findings

$ ho_{eta,eta}$-orthogonality cannot be directly compared to existing orthogonalities.

Characterization of inner product spaces using $ ho_{eta,eta}$-orthogonality.

Any linear-preserving $ ho_{eta,eta}$-orthogonality is a scalar multiple of an isometry.

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.