`BookProof.ChapterNumericalRangeCrouzeix.numRadiusLE_iff_numRange_subset` [CompleteSpace E] (A : E →L[ℂ] E) {r : ℝ} : NumRadiusLE A r ↔ BookProof.ChapterH9.numRange A ⊆ Metric.clos

Proved

BookProof.ChapterNumericalRangeCrouzeix.numRadiusLE_iff_numRange_subset

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterNumericalRangeCrouzeix.

BookProof.ChapterNumericalRangeCrouzeix.numRadiusLE_iff_numRange_subset [CompleteSpace E] (A : E →L[ℂ] E) {r : ℝ} : NumRadiusLE A r ↔ BookProof.ChapterH9.numRange A ⊆ Metric.closedBall (0 : ℂ) r

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

Preamble

-- Generated from ChapterNumericalRangeCrouzeix.lean — theorem BookProof.ChapterNumericalRangeCrouzeix.numRadiusLE_iff_numRange_subset
import Mathlib
import Definitions.Def_ChapterNumericalRangeCrouzeix
import Definitions.Def_ChapterH9
open BookProof.ChapterH9
open BookProof.ChapterNumericalRangeCrouzeix


open scoped InnerProductSpace


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

Formal statement

theorem BookProof.ChapterNumericalRangeCrouzeix.numRadiusLE_iff_numRange_subset [CompleteSpace E] (A : E →L[ℂ] E) {r : ℝ} :
    NumRadiusLE A r ↔ BookProof.ChapterH9.numRange A ⊆ Metric.closedBall (0 : ℂ) r := by sorry

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