Angles, orthogonality, and Pythagorean theorem in Banach spaces with two related applications
Year: 2026
Type: preprint
Abstract: 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.
Source: arXiv (Cornell University)
Language: English
Topic: Advanced Banach Space Theory
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Awards: G1511544777, G1622702131, Collaborative Research: Theory, computation and applications of parameterized Wasserstein gradient and Hamiltonian flows, G2389235446, Collaborative Research: Theory, computation and applications of parameterized solution operator approximation for high-dimensional PDEs, G465505792, G4772130973, G5488169599, G6155027771, G8090993009, G881421601
OpenAlex ID: W7140233439