Angles, orthogonality, and Pythagorean theorem in Banach spaces with two related applications
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866910182806650880 |
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| author | Cicone, Antonio Serra-Capizzano, Stefano Tento, Giacomo Zhou, Haomin |
| author_facet | Cicone, Antonio Serra-Capizzano, Stefano Tento, Giacomo Zhou, Haomin |
| contents | In the current work, we propose a generalization of angles and orthogonality from $L^2$ to generic Banach spaces, starting from a $L^p$ version of the Pythagorean theorem, $p\in [1,\infty)$. The starting point is conservation of energy measured in $L^1$ norm, as it occurs when considering the intrinsic mode functions decomposition in signal processing. This conservation of energy measure in $L^1$ norm is exactly the $L^1$ Pythagorean theorem. Besides the theoretical analysis, we apply the new notions in the context of preconditioning for structured large linear systems, by obtaining new classes of preconditioners. The present work contains numerical experiments and various remarks on the possible use of the given framework. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_00009 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Angles, orthogonality, and Pythagorean theorem in Banach spaces with two related applications Cicone, Antonio Serra-Capizzano, Stefano Tento, Giacomo Zhou, Haomin General Mathematics In the current work, we propose a generalization of angles and orthogonality from $L^2$ to generic Banach spaces, starting from a $L^p$ version of the Pythagorean theorem, $p\in [1,\infty)$. The starting point is conservation of energy measured in $L^1$ norm, as it occurs when considering the intrinsic mode functions decomposition in signal processing. This conservation of energy measure in $L^1$ norm is exactly the $L^1$ Pythagorean theorem. Besides the theoretical analysis, we apply the new notions in the context of preconditioning for structured large linear systems, by obtaining new classes of preconditioners. The present work contains numerical experiments and various remarks on the possible use of the given framework. |
| title | Angles, orthogonality, and Pythagorean theorem in Banach spaces with two related applications |
| topic | General Mathematics |
| url | https://arxiv.org/abs/2605.00009 |