`BookProof.ChapterNumericalRangeCrouzeix.inner_polar_bound` {A : E →L[ℂ] E} {r : ℝ} (h : NumRadiusLE A r) (x y : E) : ‖(⟪A y, x⟫_ℂ)‖ ≤ r * (‖x‖ ^ 2 + ‖y‖ ^ 2)

Proved

BookProof.ChapterNumericalRangeCrouzeix.inner_polar_bound

by leonardopedro · Oct 10, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterNumericalRangeCrouzeix.

BookProof.ChapterNumericalRangeCrouzeix.inner_polar_bound {A : E →L[ℂ] E} {r : ℝ} (h : NumRadiusLE A r) (x y : E) : ‖(⟪A y, x⟫_ℂ)‖ ≤ r * (‖x‖ ^ 2 + ‖y‖ ^ 2)

Formalization note: Lean 4 identifier BookProof.ChapterNumericalRangeCrouzeix.inner_polar_bound.

Preamble

-- Generated from ChapterNumericalRangeCrouzeix.lean — theorem BookProof.ChapterNumericalRangeCrouzeix.inner_polar_bound
import Mathlib
import Definitions.Def_ChapterNumericalRangeCrouzeix
open BookProof.ChapterNumericalRangeCrouzeix


open scoped InnerProductSpace


variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℂ E]

Formal statement

theorem BookProof.ChapterNumericalRangeCrouzeix.inner_polar_bound {A : E →L[ℂ] E} {r : ℝ} (h : NumRadiusLE A r) (x y : E) :
    ‖(⟪A y, x⟫_ℂ)‖ ≤ r * (‖x‖ ^ 2 + ‖y‖ ^ 2) := by sorry

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