Strict Positivity Near the Identity

Proved

FunctionalInequalities.KLS.LogConcave.Suspension.inner_self_pos_of_norm_sub_one_lt_readable

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

functional-inequalitieskls-conjecturekls-modularkls-readablelog-concave

This is an operator positivity estimate.

Let $E$ be a complete real inner-product normed space, let $A:E\to E$ be continuous linear, and assume $\|A-I\|<1$ . For every nonzero vector $v\in E$ ,

0<\langle Av,v\rangle.

Thus an operator lying within operator norm $1$ of the identity has a strictly positive quadratic form. The theorem is a technical criterion used to obtain invertibility from norm convergence; self-adjointness is not an explicit hypothesis in the type.

Original Lean declaration header

Within namespace FunctionalInequalities.KLS.LogConcave.Suspension. Shown in its original module context. Proof omitted.

theorem inner_self_pos_of_norm_sub_one_lt
    [CompleteSpace E]
    {A : E →L[ℝ] E} (hA : ‖A - 1‖ < 1) {v : E} (hv : v ≠ 0) :
    0 < ⟪A v, v⟫

Module navigation

Module: KLS/BKL/CompactReduction.

Previous: opNorm_le_of_inner_eq · Next: exists_standardized.

Source excerpt, lines 166–169 · Frozen source 408915f.

Preamble

import Definitions.Def_KLSModular_FunctionalInequalities_KLS_BKL_CompactReduction

open scoped RealInnerProductSpace
variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]

Formal statement


theorem FunctionalInequalities.KLS.LogConcave.Suspension.inner_self_pos_of_norm_sub_one_lt_readable
    [CompleteSpace E]
    {A : E →L[ℝ] E} (hA : ‖A - 1‖ < 1) {v : E} (hv : v ≠ 0) :
    0 < ⟪A v, v⟫ := by sorry

Source

Lean source: https://github.com/human-ai-mathematics/kls-lean/blob/408915f/formalization/FunctionalInequalities/KLS/BKL/CompactReduction.lean#L165-L181 (repository human-ai-mathematics/kls-lean, commit 408915f, module `FunctionalInequalities.KLS.BKL.CompactReduction`, declaration `FunctionalInequalities.KLS.LogConcave.Suspension.inner_self_pos_of_norm_sub_one_lt`; the repository is private at upload time).

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