Theoretical and Natural Science
- The Open Access Proceedings Series for Conferences
Theoretical and Natural Science
Vol. 30, 24 January 2024
Open
Access | Article
Beyond the finite: An exploration of infinite-dimensional vector spaces
* Author to whom correspondence should be addressed.
Abstract
In this paper, we delve deeply into the intricacies of linear algebra, with a focus on the progression from finite to infinite-dimensional vector spaces. Starting with the foundational concepts, we define vectors, vector spaces, linear combinations, and basis. The importance of infinite-dimensional vector spaces is emphasized, particularly their role in better understanding and modeling complex mathematical phenomena. Through well-illustrated examples, we guide the reader on how to validate if a given set can be classified as a vector space. Additionally, the methodology to identify bases for these vast spaces is discussed in detail. Reduction methods also play an important role in determining bases for infinite-dimensional spaces. In our conclusion, we reflect on the evolution from basic vector concepts to the more nuanced understanding of infinite dimensions. This progression not only deepens our understanding of vectors but also sets the stage for advanced investigations into linear relationships and transformations. By bridging the gap between elementary vector knowledge and advanced infinite-dimensional spaces, this paper makes a notable contribution to the ever-evolving field of linear algebra, serving as a valuable resource for both students and practitioners.
Keywords
Linear Algebra, Vector Space, Infinite Dimensions
References
1. Erwin Kreyszig, Introductory Functional Analysis with Applications, Vol. 2, 1989
2. Jörg Liesen, Volker Mehrmann, Linear Algebra, 2015, Pages 115-133
3. Muscat. J, Functional Analysis: An Introduction to Metric Spaces, Hilbert Spaces, and Banach Algebras, 2014
4. Jim Hefferon, LINEAR ALGEBRA, Vol. 4, 2020
5. Michael J. Crowe, A History of Vector Analysis: The Evolution of the Idea of a Vectorial System, Vol. 4, 1994
6. Michael A. Parker, “Vector and Hilbert Spaces”, Solid State and Quantum Theory for Optoelectronics, 2009
7. Wolfram Research, Inc., Mathematica, Version 13.3. Champaign, IL, 2023
8. Aleksandr Sergeevich Davydov, Quantum Mechanics, Mir Publishers, Elsevier Science Technology Books, 2023
9. Robert Graves Lester Telser, Functional Analysis in Mathematical Economics: Optimization over Infinite Horizons, 1972.
10. Yuka Hashimoto, T. Nodera, Krylov subspace methods for estimating operator-vector multiplications in Hilbert spaces, Journal of the Operations Research Society of Japan, 2021
Data Availability
The datasets used and/or analyzed during the current study will be available from the authors upon reasonable request.
This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License. Authors who publish this series agree to the following terms:
1. Authors retain copyright and grant the series right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgment of the work's authorship and initial publication in this series.
2. Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the series's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgment of its initial publication in this series.
3. Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See Open Access Instruction).
- Volume Title
- Proceedings of the 3rd International Conference on Computing Innovation and Applied Physics
- ISBN (Print)
- 978-1-83558-283-1
- ISBN (Online)
- 978-1-83558-284-8
- Published Date
- 24 January 2024
- Series
- Theoretical and Natural Science
- ISSN (Print)
- 2753-8818
- ISSN (Online)
- 2753-8826
- DOI
- 10.54254/2753-8818/30/20241017
- Copyright
- 24 January 2024
- Open Access
- This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited