`BookProof.ChapterNumericalRangeCrouzeix.norm_term_le` [CompleteSpace E] {A : E →L[ℂ] E} {r M R : ℝ} (hr : 0 ≤ r) (hM : 0 ≤ M) (hR : r < R) (h : NumRadiusLE A r) {a : ℕ → ℂ} (ha :

Proved

BookProof.ChapterNumericalRangeCrouzeix.norm_term_le

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterNumericalRangeCrouzeix.

BookProof.ChapterNumericalRangeCrouzeix.norm_term_le [CompleteSpace E] {A : E →L[ℂ] E} {r M R : ℝ} (hr : 0 ≤ r) (hM : 0 ≤ M) (hR : r < R) (h : NumRadiusLE A r) {a : ℕ → ℂ} (ha : ∀ n, ‖a n‖ ≤ M / R ^ n) (n : ℕ) (hn : 0 < n) : ‖a n • A ^ n‖ ≤ 2 * M * (r / R) ^ n

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

Preamble

-- Generated from ChapterNumericalRangeCrouzeix.lean — theorem BookProof.ChapterNumericalRangeCrouzeix.norm_term_le
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.norm_term_le [CompleteSpace E] {A : E →L[ℂ] E} {r M R : ℝ} (hr : 0 ≤ r)
    (hM : 0 ≤ M) (hR : r < R) (h : NumRadiusLE A r) {a : ℕ → ℂ}
    (ha : ∀ n, ‖a n‖ ≤ M / R ^ n) (n : ℕ) (hn : 0 < n) :
    ‖a n • A ^ n‖ ≤ 2 * M * (r / R) ^ n := by sorry

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