Full Text Available
Note: Clicking the button above will open the full text document at the original institutional repository in a new window.
An exploration of near-vector space theory
Thesis (PhD)--Stellenbosch University, 2026.
Saved in:
| Main Author: | |
|---|---|
| Other Authors: | |
| Format: | Thesis |
| Language: | English |
| Published: |
Stellenbosch : Stellenbosch University
2026
|
| Tags: |
No Tags, Be the first to tag this record!
|
| _version_ | 1867613736433352704 |
|---|---|
| access_status_str | Open Access |
| author | Moore, Daniella |
| author2 | Marques, Sophie |
| author_browse | Marques, Sophie Moore, Daniella |
| author_facet | Marques, Sophie Moore, Daniella |
| author_sort | Moore, Daniella |
| collection | Thesis |
| dc_rights_str_mv | Stellenbosch University |
| description | Thesis (PhD)--Stellenbosch University, 2026. |
| format | Thesis |
| id | oai:scholar.sun.ac.za:10019.1/136149 |
| institution | Stellenbosch University (South Africa) |
| language | English |
| last_indexed | 2026-06-10T12:40:53.123Z |
| license_str | Other — see source repository |
| provenance_str_mv | Harvested via OAI-PMH from SUNScholar — Stellenbosch University Repository |
| publishDate | 2026 |
| publishDateRange | 2026 |
| publishDateSort | 2026 |
| publisher | Stellenbosch : Stellenbosch University |
| publisherStr | Stellenbosch : Stellenbosch University |
| record_format | dspace |
| source_str | SUNScholar — Stellenbosch University Repository |
| spelling | oai:scholar.sun.ac.za:10019.1/136149 An exploration of near-vector space theory Moore, Daniella Marques, Sophie Janelidze, Zurab Stellenbosch University. Faculty of Science. Dept. of Mathematical Sciences. Thesis (PhD)--Stellenbosch University, 2026. Moore, D. 2026. An exploration of near-vector space theory. Unpublished doctoral dissertation. Stellenbosch: Stellenbosch University [online]. Available: https://scholar.sun.ac.za/items/03c8636e-c884-46a5-9ea3-82ac4e924bb6 There have been several definitions of near-vector spaces in the literature. In this thesis, we adopt the definition introduced by J. André. Although many properties of near-vector spaces have been established in the literature, this thesis provides algebraic proofs of several key results. One such result shows that a subspace of a near-vector space only requires the space to be nonempty and the closure under addition and scalar multiplication. Another fundamental result establishes that the quotient of a near-vector space by a subspace is itself a near-vector space. From this, we derive The First Isomorphism Theorem for near-vector spaces. Furthermore, this thesis develops the notion of linear independence and span for near-vector spaces, leading to an interesting definition of a basis for near-vector spaces. An important result proven by J. André is that any near-vector space can be decomposed into a direct sum of what is called regular near-vector spaces, a result known as The Decomposition Theorem. In some cases, these regular near-vector spaces are isomorphic, as near-vector spaces, to vector spaces. However, this is not always the case. In this thesis, we refine this result with a finer decomposition, termed The Distributive Decomposition Theorem, in which each component of the decomposition can be thought of as some vector space. A notable limitation in the theory of near-vector spaces is that there is a lack of explicit examples of near-vector spaces in the literature. To address the limitations of examples of near-vector spaces that are not vector spaces, this thesis offers a broad family of computable near-vector spaces, which we call multiplicative near-vector spaces, and study their properties. Building on these results, a natural question arises: can one define the notion of duality and show the existence of inner products for near-vector spaces? This thesis provides a definition of inner products in a broader content than the classical one, even though it does not encompass every near-vector space. This generalization gives us access to building a broad family of examples of inner products using multiplicative near-vector spaces and enables the recovery of certain classical norms that, while not defining Hilbert spaces in the classical setting, do so in our framework. Moreover, we extend the definition of generalized means to all complex numbers. Finally, the notion of near-vector spaces can be extended to André modules, which gives us access to applications and development of the theory of near-commutative algebra and the theory of near-algebraic geometry. In this thesis, it is proven that the category of André modules is abelian. Doctoral 2026-04-23T12:26:35Z 2026-04-23T12:26:35Z 2026-03 Thesis https://scholar.sun.ac.za/handle/10019.1/136149 en Stellenbosch University 150 pages application/pdf Stellenbosch : Stellenbosch University |
| spellingShingle | Moore, Daniella An exploration of near-vector space theory |
| title | An exploration of near-vector space theory |
| title_full | An exploration of near-vector space theory |
| title_fullStr | An exploration of near-vector space theory |
| title_full_unstemmed | An exploration of near-vector space theory |
| title_short | An exploration of near-vector space theory |
| title_sort | exploration of near vector space theory |
| url | https://scholar.sun.ac.za/handle/10019.1/136149 |
| work_keys_str_mv | AT mooredaniella anexplorationofnearvectorspacetheory AT mooredaniella explorationofnearvectorspacetheory |