The Complex Angle in Normed Spaces

TL;DR

This paper introduces a generalized concept of angles in complex normed spaces, exploring their properties and identifying conditions under which these angles are real, thus enabling Euclidean geometric reasoning in complex settings.

Contribution

It defines a complex-valued angle in normed spaces, analyzes its properties, and demonstrates the existence of real-valued angles in complex and inner product spaces.

Findings

Complex angles can be real-valued in certain complex normed spaces.

Inner product spaces contain many pairs of vectors with real angles.

The theory enables Euclidean geometry in complex vector spaces.

Abstract

We consider a generalized angle in complex normed vector spaces. Its definition corresponds to the definition of the well known Euclidean angle in real inner product spaces. Not surprisingly it yields complex values as `angles'. This `angle' has some simple properties, which are known from the usual angle in real inner product spaces. But to do ordinary Euclidean geometry real angles are necessary. We show that even in a complex normed space there are many pure real valued `angles'. The situation improves yet in inner product spaces. There we can use the known theory of orthogonal systems to find many pairs of vectors with real angles, and to do geometry which is based on the Greeks 2000 years ago.