On Uniqueness of New Orthogonality via 2-HH Norm in Normed Linear Space

Bhuwan Prasad Ojha,Vishnu Narayan Mishra,Prakash Muni Bajracharya,Syed Abdul Mohiuddine

doi:10.1155/2020/8835492

Bhuwan Prasad Ojha, Vishnu Narayan Mishra + Show 2 more

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https://doi.org/10.1155/2020/8835492

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Abstract

This paper generalizes the special case of the Carlsson orthogonality in terms of the 2-HH norm in real normed linear space. Dragomir and Kikianty (2010) proved in their paper that the Pythagorean orthogonality is unique in any normed linear space, and isosceles orthogonality is unique if and only if the space is strictly convex. This paper deals with the complete proof of the uniqueness of the new orthogonality through the medium of the 2-HH norm. We also proved that the Birkhoff and Robert orthogonality via the 2-HH norm are equivalent, whenever the underlying space is a real inner-product space.

Highlights

Different notions of orthogonality in normed linear spaces have been developed by various mathematicians
The property of the uniqueness of isosceles orthogonality was not discussed until Kapoor and Prasad’s paper was published. They proved that the Pythagorean orthogonality is unique in any normed linear space; the isosceles orthogonality is unique if and only if the space is strictly convex [5]
They proved that the Pythagorean orthogonality via 2-HH norm satisfies the nondegeneracy, continuity, and symmetry properties; it is neither additive nor Journal of Function Spaces homogeneous in normed linear space, but it satisfies the property of existence and uniqueness in any normed linear spaces

Summary

Introduction

Different notions of orthogonality in normed linear spaces have been developed by various mathematicians. Using the concept of the p-HH norm as described in the paper [9], Kikianty and Dragomir came up with a new notion of orthogonality with the help of the 2-HH norm, which is closely related to the Pythagorean and isosceles orthogonalities [10]. They proved that the Pythagorean orthogonality via 2-HH norm satisfies the nondegeneracy, continuity, and symmetry properties; it is neither additive nor Journal of Function Spaces homogeneous in normed linear space, but it satisfies the property of existence and uniqueness in any normed linear spaces. If the norm on X is induced by an inner product, the Robert and Birkhoff orthogonality via the 2-HH norm is equivalent

Definition Notation and Preliminary Results

Main Result