Squared Extended Norm of a Finite Sum

Proved

LevyStochCalc.Brownian.Ito.sq_enorm_finsetSum_le_readable

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

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

For m : ℕ and a real-valued family x : Fin m → ℝ, the squared extended norm of its finite sum is bounded in ENNReal by m times the sum of the squared extended norms:

\left(\left\|\sum_{k<m}x_k\right\|_+\!:\!\mathbb{R}_{\ge0\infty}\right)^2\le m\sum_{k<m}\left(\|x_k\|_+\!:\!\mathbb{R}_{\ge0\infty}\right)^2.

The lemma sq_enorm_finsetSum_le_readable is a finite-dimensional quadratic estimate in the extended nonnegative reals.

Original Lean declaration header

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

theorem sq_enorm_finsetSum_le {m : ℕ} (x : Fin m → ℝ) :
    (‖∑ k, x k‖₊ : ℝ≥0∞) ^ 2 ≤ (m : ℝ≥0∞) * ∑ k, (‖x k‖₊ : ℝ≥0∞) ^ 2

Module navigation

Module: LevyStochCalc/Brownian/ItoTruncate.

Previous: tendsto_energy_clampAt_sub.

Source excerpt, lines 178–179 · Frozen source 408915f.

Preamble

import Definitions.Def_KLSModular_LevyStochCalc_Brownian_ItoTruncate

/-
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/ItoTruncate.lean; Apache-2.0; provenance in formalization/vendor/vendor-lock.json of the KLS repository)
open scoped NNReal ENNReal Topology
universe u
variable {Ω : Type u} [MeasurableSpace Ω] {P : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure P]

Formal statement


theorem LevyStochCalc.Brownian.Ito.sq_enorm_finsetSum_le_readable {m : ℕ} (x : Fin m → ℝ) :
    (‖∑ k, x k‖₊ : ℝ≥0∞) ^ 2 ≤ (m : ℝ≥0∞) * ∑ k, (‖x k‖₊ : ℝ≥0∞) ^ 2 := by sorry

Source

Lean source: https://github.com/cgarryZA/LevyStochCalc/blob/984dbf0b790031eac05d56fe5a84b65e15e1e932/LevyStochCalc/Brownian/ItoTruncate.lean#L176-L195 (repository human-ai-mathematics/kls-lean, commit 408915f, module `LevyStochCalc.Brownian.ItoTruncate`, declaration `LevyStochCalc.Brownian.Ito.sq_enorm_finsetSum_le`; 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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