Semester
Summer
Date of Graduation
2021
Document Type
Dissertation
Degree Type
PhD
College
Eberly College of Arts and Sciences
Department
Mathematics
Committee Chair
Harry Gingold
Committee Co-Chair
Jocelyn Quaintance
Committee Member
Dening Li
Committee Member
Harumi Hattori
Committee Member
Jerzy Wojciechowski
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 an inner product vector space over R. Using this arrow space pre-inner product, various properties of the arrow space are derived and contrasted with the properties of a vector space over R. 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. With arrow space's tools, Hilbert's axioms of Euclidean plane geometry follow as theorems in arrow spaces. Applications of using an arrow space to solve geometric problems in affine geometry are provided. Examples are provided to equip complex Hilbert spaces with a structure analogous to the structure of Euclidean geometry and trigonometry.
Recommended Citation
ALBAHBOH, HUSSIN M., "Arrow Spaces: A Unified Algebraic Approach to Euclidean Geometry and Inner Product Spaces" (2021). Graduate Theses, Dissertations, and Problem Reports. 8307.
https://researchrepository.wvu.edu/etd/8307