Bilinear Operator Norm Bound by Coordinates

Proved

LevyStochCalc.norm_le_sum_abs_apply_two_readable

by Lazyfill · Oct 26, 2026 · Mathlib 0df444a (Lean v4.33.1)

functional-inequalitieskls-conjecturekls-modularkls-readablelog-concavevendored-levystochcalc

This bounds a continuous bilinear form by its coordinate values.

For $n\in\mathbb N$ and a continuous linear map

A:(\operatorname{Fin}(n)\to\mathbb R)\to \bigl((\operatorname{Fin}(n)\to\mathbb R)\to\mathbb R\bigr),

one has

\|A\|\le \sum_{p\in\operatorname{Fin}(n)}\sum_{q\in\operatorname{Fin}(n)} \left|A(e_p)(e_q)\right|,

where $e_p$ and $e_q$ are standard basis vectors.

This provides a coordinate bound for second derivatives and bilinear terms.

Original Lean declaration header

Within namespace LevyStochCalc. Shown in its original module context. Proof omitted.

theorem norm_le_sum_abs_apply₂ (A : (Fin n → ℝ) →L[ℝ] (Fin n → ℝ) →L[ℝ] ℝ) :
    ‖A‖ ≤ ∑ p, ∑ q, |A (Pi.single p 1) (Pi.single q 1)|

Module navigation

Module: LevyStochCalc/Brownian/CoordDerivative.

Previous: norm_le_sum_abs_apply · Next: abs_coordDeriv₂_sub_le_affine.

Source excerpt, lines 163–164 · Frozen source 408915f.

Preamble

import Definitions.Def_KLSModular_LevyStochCalc_Brownian_CoordDerivative

/-
Copyright (c) 2026 Christian Garry. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christian Garry
-/
-- Vendored from https://github.com/cgarryZA/LevyStochCalc at commit 984dbf0b790031eac05d56fe5a84b65e15e1e932 (upstream path LevyStochCalc/Brownian/CoordDerivative.lean; Apache-2.0; provenance in formalization/vendor/vendor-lock.json of the KLS repository)
variable {n : ℕ}

Formal statement


theorem LevyStochCalc.norm_le_sum_abs_apply_two_readable (A : (Fin n → ℝ) →L[ℝ] (Fin n → ℝ) →L[ℝ] ℝ) :
    ‖A‖ ≤ ∑ p, ∑ q, |A (Pi.single p 1) (Pi.single q 1)| := by sorry

Source

Lean source: https://github.com/cgarryZA/LevyStochCalc/blob/984dbf0b790031eac05d56fe5a84b65e15e1e932/LevyStochCalc/Brownian/CoordDerivative.lean#L161-L202 (repository human-ai-mathematics/kls-lean, commit 408915f, module `LevyStochCalc.Brownian.CoordDerivative`, declaration `LevyStochCalc.norm_le_sum_abs_apply₂`; the repository is private at upload time). Vendored from https://github.com/cgarryZA/LevyStochCalc at commit 984dbf0b790031eac05d56fe5a84b65e15e1e932 (Apache-2.0; upstream copyright notice retained in the Lean text).

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