Arrow Spaces: An Approach to Inner Product Spaces and Affine Geometry

TL;DR

This paper introduces the concept of arrow spaces with a pre-inner product, establishing axioms that lead to vector space properties and applying this framework to solve geometric problems in affine geometry.

Contribution

It proposes a new axiomatic framework for arrow spaces with a pre-inner product, deriving vector space axioms and demonstrating applications in affine geometry.

Findings

Arrow spaces can be axiomatized with a pre-inner product.

Vector space axioms follow from arrow space axioms.

Applications to affine geometry problems are demonstrated.

Abstract

Given a postulated set of points, an algebraic system of axioms is proposed for an "arrow space'". An arrow is defined to be an ordered set of two points <T, H>, named respectively Tail and Head. The set of arrows is an arrow space. The arrow space is axiomatically endowed with an arrow space "pre-inner product" which is analogous to the inner product of a Euclidean vector space. Using this arrow space pre-inner product, various properties of the arrow space are derived and contrasted with the properties of a Euclidean vector space. The axioms of a vector space and its associated inner product are derived as theorems that follow from the axioms of an arrow space since vectors are rigorously shown to be equivalence classes of arrows. Applications of using an arrow space to solve geometric problems in affine geometry are provided.