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arxiv: 2605.00009 · v1 · submitted 2026-02-28 · 🧮 math.GM

Angles, orthogonality, and Pythagorean theorem in Banach spaces with two related applications

Antonio Cicone , Stefano Serra-Capizzano , Giacomo Tento , Haomin Zhou This is my paper

Pith reviewed 2026-05-15 18:45 UTC · model grok-4.3

classification 🧮 math.GM
keywords Banach spacesPythagorean theoremanglesorthogonalitypreconditionersLp normssignal processinglinear systems
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The pith

Angles and orthogonality extend to any Banach space via an Lp version of the Pythagorean theorem for p from 1 to infinity.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper defines angles and orthogonality in general Banach spaces by generalizing the Pythagorean theorem to Lp spaces. It takes the conservation of energy in the L1 norm, observed in intrinsic mode function decompositions for signals, as the L1 case of that theorem. The resulting notions recover the familiar inner-product definitions when the space is L2. The same framework produces new preconditioners for large structured linear systems and is tested numerically in that setting.

Core claim

We propose a generalization of angles and orthogonality from L² 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 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.

What carries the argument

The Lp Pythagorean theorem for p in [1, infinity), used to define the cosine of the angle between two vectors so that the norm of their sum equals the sum of their norms precisely when they are orthogonal.

If this is right

  • The new angle reduces exactly to the classical angle when the underlying space is L2.
  • Generalized orthogonality permits decompositions and projections with respect to arbitrary Lp norms.
  • Preconditioners constructed from these angles improve iterative solvers for structured linear systems.
  • Numerical tests confirm that the new preconditioners perform as predicted in practice.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The construction supplies a uniform way to measure angles under any norm, opening the possibility of norm-specific Fourier or wavelet bases.
  • In finite dimensions the same definition gives angles with respect to the l1 or l-infinity norm that could be checked directly by enumeration of basis vectors.
  • The approach may extend variational principles or optimization algorithms that currently assume Hilbert-space geometry.

Load-bearing premise

That energy conservation in the L1 norm from intrinsic mode function decomposition directly supplies a Pythagorean relation that defines angles and orthogonality in every Banach space.

What would settle it

A pair of vectors in a concrete Banach space such as L1 or l-infinity where the proposed angle yields a value that violates the Lp Pythagorean equality or fails to reduce to the standard arccos inner-product formula at p=2.

Figures

Figures reproduced from arXiv: 2605.00009 by Antonio Cicone, Giacomo Tento, Haomin Zhou, Stefano Serra-Capizzano.

Figure 1
Figure 1. Figure 1: Left panel, the signal s and its mono-frequency non-stationary components, first row and subsequent ones, respectively. Central panel, FIF decomposition using default parameter values (α = 30, and Ξ = 1.6). Right panel, FIF decomposition with α = 100, and Ξ = 3. We can observe from Figures 2 and 3 that both decompositions conserve, up to machine precision, the L 1 Fourier Energy of the signal, see right pa… view at source ↗
Figure 2
Figure 2. Figure 2: Left panel, the discrete Fourier transform absolute valu [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Left panel, |sb(ξk)| and n |IMF d j (ξk)| o of the IMFs produced by FIF with parameter values α = 100, and Ξ = 3, and displayed in the central panel of [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The paper proposes a generalization of angles and orthogonality from L² to generic Banach spaces, starting from an L^p version of the Pythagorean theorem for p ∈ [1, ∞). The foundation is the claim that energy conservation in the L¹ norm during intrinsic mode function (IMF) decomposition in signal processing constitutes an exact L¹ Pythagorean theorem, which is then extended to define angles and orthogonality in arbitrary Banach spaces. The work includes theoretical analysis, applications to preconditioning for structured large linear systems yielding new preconditioner classes, and numerical experiments.

Significance. If rigorously established, the framework could provide a useful geometric tool for Banach spaces lacking inner-product structure, with potential applications in signal processing and numerical linear algebra. The preconditioning application and experiments offer concrete illustrations, but the significance depends on whether the L¹-to-L^p extension and angle definition satisfy the necessary algebraic consistency.

major comments (3)
  1. [Abstract / Theoretical analysis] Abstract and opening theoretical section: The assertion that L¹ energy conservation in IMF decomposition 'is exactly the L¹ Pythagorean theorem' (i.e., ||x + y||₁ = ||x||₁ + ||y||₁ under IMF-induced orthogonality) is load-bearing but unsupported; IMF/EMD typically yields a nonzero residue, and exact L¹ additivity holds only for functions with disjoint supports, not as a general consequence of the decomposition.
  2. [Theoretical analysis] Theoretical analysis: The extension from the L¹ case to a general L^p Pythagorean theorem for arbitrary p ∈ [1, ∞) and its use to define angles/orthogonality in generic Banach spaces lacks a derivation showing consistency with existing notions such as Birkhoff-James orthogonality; the proposed angle (presumably via normalized L^p difference) must be shown to possess the required properties (e.g., symmetry, range in [0, π/2]) for the claim to hold.
  3. [Preconditioning application] Preconditioning application: The derivation of new preconditioner classes from the generalized orthogonality rests on the unverified L^p extension; without an explicit proof that the angle definition induces a valid orthogonality relation compatible with the Banach space norm, the application claims cannot be substantiated.
minor comments (2)
  1. [Numerical experiments] Numerical experiments: Provide explicit error metrics, matrix dimensions, and baseline comparisons (e.g., standard ILU or diagonal preconditioners) to allow assessment of the reported improvements.
  2. [Throughout] Notation: Clarify the exact formula used for the generalized angle (e.g., the normalized L^p expression) and ensure it is stated uniformly across sections.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address each major point below, providing clarifications and indicating where revisions will be made to improve rigor and consistency.

read point-by-point responses

  1. Referee: [Abstract / Theoretical analysis] Abstract and opening theoretical section: The assertion that L¹ energy conservation in IMF decomposition 'is exactly the L¹ Pythagorean theorem' (i.e., ||x + y||₁ = ||x||₁ + ||y||₁ under IMF-induced orthogonality) is load-bearing but unsupported; IMF/EMD typically yields a nonzero residue, and exact L¹ additivity holds only for functions with disjoint supports, not as a general consequence of the decomposition.

    Authors: We agree that standard EMD often includes a residue term. In the manuscript we restrict attention to the complete-decomposition case in which the sum of the extracted IMFs exactly recovers the original signal (a setting frequently adopted in theoretical studies of EMD orthogonality). Under this assumption the observed L¹ energy conservation is precisely the statement ||x + y||₁ = ||x||₁ + ||y||₁. We will revise the abstract and the opening theoretical section to state this assumption explicitly and to note that exact additivity holds when the IMFs have essentially disjoint supports (or satisfy the minimal-overlap condition that produces the conservation property). A short remark linking the IMF notion to the classical disjoint-support case will be added. revision: partial

  2. Referee: [Theoretical analysis] Theoretical analysis: The extension from the L¹ case to a general L^p Pythagorean theorem for arbitrary p ∈ [1, ∞) and its use to define angles/orthogonality in generic Banach spaces lacks a derivation showing consistency with existing notions such as Birkhoff-James orthogonality; the proposed angle (presumably via normalized L^p difference) must be shown to possess the required properties (e.g., symmetry, range in [0, π/2]) for the claim to hold.

    Authors: We will insert an explicit derivation subsection. Starting from the proposed L^p Pythagorean relation ||x + y||_p^p = ||x||_p^p + ||y||_p^p, the angle is defined by the normalized expression cos θ = (||x||_p^p + ||y||_p^p − ||x + y||_p^p) / (2 ||x||_p ||y||_p). We verify that this reduces to the classical inner-product angle when p = 2, is symmetric in x and y, and takes values in [0, π/2]. A direct comparison with Birkhoff-James orthogonality (||x + λ y|| ≥ ||x|| for all real λ) is provided, showing that our relation implies the Birkhoff-James condition for p = 2 and yields a natural p-norm analogue for p ≠ 2. These properties and the short proof will be added to the theoretical analysis section. revision: yes

  3. Referee: [Preconditioning application] Preconditioning application: The derivation of new preconditioner classes from the generalized orthogonality rests on the unverified L^p extension; without an explicit proof that the angle definition induces a valid orthogonality relation compatible with the Banach space norm, the application claims cannot be substantiated.

    Authors: We will add a dedicated paragraph deriving the preconditioner construction directly from the angle/orthogonality definition. The argument shows that the generalized orthogonality relation is compatible with the underlying Banach norm (satisfies the triangle inequality and yields a well-defined splitting of the operator into “orthogonal” components). The resulting preconditioner matrices are then expressed in closed form using the angle formula. This explicit link, together with the already-present numerical experiments, will substantiate the application claims. revision: yes

Circularity Check

1 steps flagged

L1 conservation from IMF presented as exact L1 Pythagorean theorem without independent derivation

specific steps
  1. self definitional [Abstract]
    "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."

    The text directly equates the L1-norm conservation property observed in IMF decomposition to the L1 Pythagorean theorem by definition, making the theorem an input taken from the signal-processing context rather than independently derived; this identification then serves as the foundation for generalizing angles and orthogonality to arbitrary Banach spaces.

full rationale

The paper's derivation chain begins by identifying observed L1-norm conservation during IMF decomposition as the L1 Pythagorean theorem itself, then uses this to motivate an Lp generalization and angle definitions in general Banach spaces. This identification is load-bearing for the subsequent claims but rests on equating an empirical signal-processing property to a mathematical theorem without a separate axiomatic or proof-based foundation shown in the provided text. No self-citation chain or explicit fitting of parameters to the target result is evident, and the central generalization step retains independent content beyond the initial identification. The preconditioning application and experiments therefore inherit this foundational step but do not reduce the entire result to a tautology.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on treating L1 energy conservation from signal processing as the basis for an L1 Pythagorean theorem that then defines angles in Banach spaces.

axioms (1)
  • domain assumption L1 norm conservation of energy holds for intrinsic mode function decompositions and directly implies an L1 Pythagorean theorem
    Invoked as the starting point for the entire generalization

pith-pipeline@v0.9.0 · 5430 in / 1163 out tokens · 42630 ms · 2026-05-15T18:45:25.250729+00:00 · methodology

discussion (0)

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Reference graph

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