On Birkhoff angles in normed spaces

TL;DR

This paper explores Birkhoff angles in normed spaces, analyzing their properties, how to compare their acuteness or obtuseness, and providing formulas and examples for computing these angles.

Contribution

It introduces a detailed study of Birkhoff angles, including properties, comparison methods, and formulas for their computation in normed spaces.

Findings

Birkhoff angles have well-defined basic properties.

Methods to compare angles' acuteness or obtuseness are established.

Formulas for computing Birkhoff 'cosine' and angles are provided with examples.

Abstract

Associated to Birkhoff orthogonality, we study Birkhoff angles in a normed space and present some of their basic properties. We also discuss how to decide whether an angle is more acute or more obtuse than another. In addition, given two vectors $x$ and $y$ in a normed space, we study the formula for Birkhoff `cosine' of the angle from $x$ to $y$ from which we can, in principal, compute the angle. Some examples will be presented.