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Angles, orthogonality, and Pythagorean theorem in Banach spaces with two related applications
Abstract
In the current work, we propose a generalization of angles and orthogonality from L2 to generic Banach spaces, starting from a Lp version of the Pythagorean theorem, p∈ [1,∞). The starting point is conservation of energy measured in L1 norm, as it occurs when considering the intrinsic mode functions decomposition in signal processing. This conservation of energy measure in L1 norm is exactly the L1 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.
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BibTeX
@misc{cicone2026angles,
title = {{Angles, orthogonality, and Pythagorean theorem in Banach spaces with two related applications}},
author = {Antonio Cicone and Stefano Serra-Capizzano and Giacomo Tento and Haomin Zhou},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2605.00009},
eprint = {2605.00009},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.00009},
}
RIS
TY - UNPB
TI - Angles, orthogonality, and Pythagorean theorem in Banach spaces with two related applications
AU - Antonio Cicone
AU - Stefano Serra-Capizzano
AU - Giacomo Tento
AU - Haomin Zhou
PY - 2026
DA - 2026/09/08
PB - arXiv
DO - 10.48550/arxiv.2605.00009
UR - https://arxiv.org/abs/2605.00009
AB - 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∈ [1,∞). 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.
ER -