Inner products for strongly regular near-vector spaces and duality for finite dimensional near-vector spaces
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866917080897421312 |
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| author | Boonzaaier, Leeandro Marques, Sophie Moore, Daniella |
| author_facet | Boonzaaier, Leeandro Marques, Sophie Moore, Daniella |
| contents | In this paper we develop a duality theory for all finite-dimensional near-vector spaces and introduce a notion of inner product tailored to the broad and natural class of strongly regular near-vector spaces. This generalized construction extends the classical inner product beyond the classical framework, yielding rich families of examples on multiplicative near-vector spaces. Within this setting, several familiar norms-such as those that fail to produce Hilbert spaces in the classical sense-emerge naturally as genuine inner-product-type norms. A further contribution is the extension of the theory of generalized (weighted) means to arbitrary complex datasets. This extension unifies and generalizes the classical power and geometric means, carrying them beyond the domain of positive reals. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_11774 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Inner products for strongly regular near-vector spaces and duality for finite dimensional near-vector spaces Boonzaaier, Leeandro Marques, Sophie Moore, Daniella General Mathematics In this paper we develop a duality theory for all finite-dimensional near-vector spaces and introduce a notion of inner product tailored to the broad and natural class of strongly regular near-vector spaces. This generalized construction extends the classical inner product beyond the classical framework, yielding rich families of examples on multiplicative near-vector spaces. Within this setting, several familiar norms-such as those that fail to produce Hilbert spaces in the classical sense-emerge naturally as genuine inner-product-type norms. A further contribution is the extension of the theory of generalized (weighted) means to arbitrary complex datasets. This extension unifies and generalizes the classical power and geometric means, carrying them beyond the domain of positive reals. |
| title | Inner products for strongly regular near-vector spaces and duality for finite dimensional near-vector spaces |
| topic | General Mathematics |
| url | https://arxiv.org/abs/2511.11774 |