Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On approximate Birkhoff-James orthogonality and normal cones in a normed space

Debmalya Sain, Kallol Paul, Arpita mal

Abstract

We study two notions of approximate Birkhoff-James orthogonality in a normed space, from a geometric point of view, and characterize them in terms of normal cones. We further explore the interconnection between normal cones and approximate Birkhoff-James orthogonality to obtain a complete characterization of normal cones in a two-dimensional smooth Banach space. We also obtain a uniqueness theorem for approximate Birkhoff-James orthogonality set in a normed space.

On approximate Birkhoff-James orthogonality and normal cones in a normed space

Abstract

We study two notions of approximate Birkhoff-James orthogonality in a normed space, from a geometric point of view, and characterize them in terms of normal cones. We further explore the interconnection between normal cones and approximate Birkhoff-James orthogonality to obtain a complete characterization of normal cones in a two-dimensional smooth Banach space. We also obtain a uniqueness theorem for approximate Birkhoff-James orthogonality set in a normed space.

Paper Structure

This paper contains 2 sections, 11 theorems, 25 equations, 2 figures.

Table of Contents

  1. Introduction.
  2. Main results.

Key Result

Theorem 1.1

Let $\mathbb{X}$ be a real normed space. For $x, y \in \mathbb{X}$ and $\epsilon \in [0,1) :$

Figures (2)

  • Figure 1:
  • Figure 2:

Theorems & Definitions (25)

  • Theorem 1.1: Theorem 2.3,CSW
  • Theorem 2.1
  • proof
  • Remark 2.1
  • Remark 2.2
  • Theorem 2.2
  • proof
  • Theorem 2.3
  • Theorem 2.4
  • proof
  • ...and 15 more