Best approximation from the positive cone of an inner product lattice
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866908494691565568 |
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| author | Abdouli, Marouen Boulabiar, Karim |
| author_facet | Abdouli, Marouen Boulabiar, Karim |
| contents | Let $E$ be a directed (i.e., positively generated) ordered vector space endowed with an inner product. In this note, we prove that the following statements are equivalent:
i) $E$ is a vector lattice and its norm induced by its inner product is a lattice norm.
ii) The metric projection onto the positive cone $E^+$ of $E$ exists and it is both isotone and subadditive.
Moreover, in this case, the best approximation to any $x \in E$ from $E^+$ coincides with its positive part $x^+$. This result extends previous work to the non-complete setting. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_13722 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Best approximation from the positive cone of an inner product lattice Abdouli, Marouen Boulabiar, Karim Functional Analysis Let $E$ be a directed (i.e., positively generated) ordered vector space endowed with an inner product. In this note, we prove that the following statements are equivalent: i) $E$ is a vector lattice and its norm induced by its inner product is a lattice norm. ii) The metric projection onto the positive cone $E^+$ of $E$ exists and it is both isotone and subadditive. Moreover, in this case, the best approximation to any $x \in E$ from $E^+$ coincides with its positive part $x^+$. This result extends previous work to the non-complete setting. |
| title | Best approximation from the positive cone of an inner product lattice |
| topic | Functional Analysis |
| url | https://arxiv.org/abs/2508.13722 |